Can a quadratic equation have more than two solutions?

No, a quadratic equation can have at most two solutions, which can be real or complex.

What happens when a quadratic equation has a negative discriminant?

A negative discriminant means the equation has no real roots, and the solutions are complex (imaginary).

Can I always factor a quadratic equation?

Not all quadratic equations can be easily factored. Some must be solved by completing the square or using the quadratic formula.

What are the roots or solutions of a quadratic equation?

The roots are the values of x where the quadratic equation equals zero and can be determined through graphing, factoring, or other means.

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Quadratic Formula

1.0Master the Quadratic Formula in Minutes

Class: 10 Mathematics (CBSE)
Chapter: Quadratic Equations (Subtopic: Quadratic Formula)
Estimated Learning Time: 20–25 Minutes

Not every quadratic equation can be factorized easily, but every quadratic equation can be solved using the Quadratic Formula. Learn how this universal formula helps calculate the roots of any quadratic equation, understand the role of the discriminant in identifying the nature of roots, and apply the method to solve real-life mathematical problems involving dimensions, speed, and motion. With step-by-step explanations and board-focused examples, this topic strengthens your algebraic problem-solving skills.

2.0Learning Outcomes

After completing this topic, you will be able to:

Identify a quadratic equation in standard form.
Apply the quadratic formula to solve equations.
Understand the meaning of the discriminant.
Determine whether the roots are real, equal, or imaginary.
Solve application-based problems using the quadratic formula.
Compare factorization and quadratic formula methods.
Solve NCERT, competency-based, and CBSE Board examination questions confidently.

3.0Introduction to the Quadratic Formula

The Quadratic Formula is one of the most powerful methods for solving quadratic equations. While some equations can be solved easily by factorization, many cannot be factorized into simple integers. In such cases, the quadratic formula provides a direct and reliable method to find the roots of any quadratic equation.

In this topic, you'll learn how to derive and apply the quadratic formula, determine the nature of roots using the discriminant, and solve board-level numerical and application-based questions with confidence. This formula is an essential part of Class 10 Mathematics and frequently appears in CBSE Board examinations.

A quadratic equation, therefore, refers to a mathematical relationship between a variable and the constants, where the highest power of the variable is squared. Sometimes, solving quadratic equations can be difficult, especially when factoring does not work or is not done. One of the best methods available for their solution is the quadratic formula, the famous formula which gives the solutions directly.

4.0General Form of Quadratic Equation

The General Form of Quadratic Equation is an important aspect of solving quadratic equations as it allows systematic methods for finding the solutions of the equation. The standard form of quadratic equation can be written as:

$a x^{2} + b x + c = 0$

Here,

x is the variable or unknown we are solving for,
a, b, and c are constants known as the coefficients, and
$a \neq = 0$ (if a = 0, the equation would be linear, not quadratic, because the highest power of x would then be 1)

5.0Quadratic Equation Formula

The quadratic equation formula, also known as the quadratic formula, is used for finding the roots of the equation in a standard form, which is $(a x^{2} + b x + c = 0)$ . It provides solutions or roots of the equation without a need to use other tougher methods. The Quadratic Formula can be expressed as:

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Here, “ $(b^{2} - 4 a c)$ ” is known as the discriminant, which can be symbolised as “D”. It helps in determining the nature of roots:

If D＞0, then the roots of the equation will have two real “distinct” roots.
If D=0, then the equation will have two “equal” roots.
If D＜0, then the roots of the equation will be imaginary.

6.0How to Solve Quadratic Equations by Quadratic Formula

Solving quadratic equations can become easy by using the following four steps:

Step 1: Identify the values of a, b, and c from the quadratic equation.
Step 2: Put these values into the quadratic formula.
Step 3: Reduce the expression under the square root and determine whether the discriminant is positive, zero, or negative. The two former cases translate to two real solutions and one real solution, respectively. The latter gives complex solutions.
Step 4: Solve for x.

7.0Derivation of Quadratic Equation

The formula for the quadratic equation can be derived by using the standard form of the quadratic equation. Like this:

$a x^{2} + b x + c = 0$

Divide the equation by the “a”

$x^{2} + (\frac{b}{a}) x + (\frac{c}{a}) = 0$ …..(1)

Using the completing the square method, taking the coefficient of x, dividing it by ½, and then squaring it:

$\frac{b}{a} \Rightarrow \frac{b}{2 a} \Rightarrow \frac{b ^{2}}{4 a ^{2}}$

Adding and subtracting $\frac{b ^{2}}{4 a ^{2}}$ on both sides in equation (1)

$x^{2} + (\frac{b}{a}) x + \frac{b ^{2}}{4 a ^{2}} + (\frac{c}{a}) - \frac{b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} + \frac{c}{a} - \frac{b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} + \frac{4 a c - b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} = \frac{- ( 4 a c - b ^{2} )}{4 a ^{2}}$

$(x + \frac{b}{2 a}) = \pm \frac{b ^{2} - 4 a c}{4 a ^{2}}$

$x = - \frac{b}{2 a} \pm \frac{b ^{2} - 4 a c}{2 a}$

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

8.0Graphing Quadratic Equations

Graphing quadratic equations is a graphical method of finding the solutions, or roots, of a quadratic equation and understanding the behaviour of the equation written in the standard form $(a x^{2} + b x + c = 0)$ . This equation produces a parabola, which is a symmetrical curve. The shape of the parabola and its orientation (whether it opens upwards or downwards) depends on the value of a.

Steps for Graphing Quadratic Equations

Write the quadratic equation in its standard form.
Find the vertex of x - coordinate by using the formula $(x = \frac{- b}{2 a})$ and substituting the values of a and b. To find the y coordinate, simply put this value of the x-coordinate in the quadratic equation.
Find the axis of symmetry, which is a vertical passing through the vertex by using again the formula of $(x = \frac{- b}{2 a})$
Evaluate y when x = 0 and the y-intercept is (0, c) to find the y-intercept.
Draw the parabola by smoothly connecting the points, ensuring the curve opens upward (if a > 0) or downward (if a < 0).
The point where the parabola intercepts on the x-axis is the root of the quadratic equations.

9.0Quadratic Equation Examples

Problem 1: Find the roots of x²+ 6x − 7 = 0 using the quadratic formula.

Solution: Given that, x²+ 6x − 7 = 0

a = 1, b = 6, c = –7

$D = b^{2} - 4 a c = 6^{2} - 4 \times 1 \times (- 7)$

$D = 36 + 28 = 64$

$x = \frac{- 6 \pm 64}{2 ( 1 )} = \frac{- 6 \pm 8}{2}$

$x = \frac{- 6 + 8}{2}, x = \frac{- 6 - 8}{2}$

$x = 1, - 7$

Problem 2: Determine if the given equation x²+ 2x + 5 = 0 has a real root or not, if the roots are real, then find the roots of the equation.

Solution: Given that x²+ 2x + 5 = 0

a = 1, b = 2, c = 5

$D = b^{2} - 4 a c = 2^{2} - 4 (1 \times 5) = 4 - 20$

D = –16

Hence, the root of the given equation is imaginary.

Problem 3: The roots of the following equation x²− bx + 9 = 0 are equal, which is 3. Find the value of b.

Solution: Given that x²− bx + 9 = 0

a = 1, c = 9, b = ?

As the roots of the equations are equal hence, D = 0

$D = b^{2} - 4 a c$

$0 = b^{2} - 4 \times 1 \times 9$

$b^{2} = 36$

$b = 36 = 6$

10.0EUREKA by ALLEN – The Future of Class 10 Learning

EUREKA by ALLEN is an advanced online learning platform designed for Class 10 students preparing for CBSE and State Board exams. With interactive story-based lessons, expert faculty guidance, AI-powered doubt-solving, and board-focused practice, EUREKA helps students strengthen concepts, improve retention, and score higher in class 10 board examinations.

Key Features of EUREKA Class 10 Courses:
Easy-to-follow concept explanations Story-based learning for better retention Exam-oriented preparation strategy Answer writing practice sessions Instant quizzes and self-assessment Performance tracking dashboard Round-the-clock AI doubt support NCERT-based curriculum coverage Study at your own pace

Explore Now

11.0Supporting Study Materials

These CBSE notes and NCERT solutions for the chapter “Polynomials”, covering topics like Quadratic Formulas, are prepared in accordance with the latest CBSE Class 10 Maths syllabus and NCERT guidelines. The study material offers detailed explanations of important concepts, definitions, examples, and exam-orientated questions to help students understand redox reactions, corrosion, and rancidity effectively while strengthening their preparation for board examinations.

CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations	NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations

12.0Previous Year Questions (PYQs) on Quadratic Formula

Question: A quadratic equation has a = 1, b = –8, and c = 16. Determine the nature of its roots.

Answer

Use the discriminant:

Δ = (–8)² – 4(1)(16)

= 64 – 64

= 0

Therefore, The equation has two equal real roots.

Question: Find two consecutive integers whose product is 210.

Answer: Let the integers be x and x + 1.

x(x + 1) = 210

x² + x – 210 = 0

Using the quadratic formula,

x = 14 Therefore, The integers are 14 and 15.

30-Second Quick Review: Quadratic Formula

Standard form: ax² + bx + c = 0, where a ≠ 0
The Quadratic Formula can solve every quadratic equation.
Discriminant: Δ = b² – 4ac
If Δ > 0 → Two distinct real roots.
If Δ = 0 → Two equal real roots.
If Δ < 0 → No real roots.
The ± sign gives two possible solutions.
The quadratic formula is especially useful when factorization is difficult or impossible.

Quadratic Formula

1.0Master the Quadratic Formula in Minutes

Class: 10 Mathematics (CBSE)
Chapter: Quadratic Equations (Subtopic: Quadratic Formula)
Estimated Learning Time: 20–25 Minutes

Not every quadratic equation can be factorized easily, but every quadratic equation can be solved using the Quadratic Formula. Learn how this universal formula helps calculate the roots of any quadratic equation, understand the role of the discriminant in identifying the nature of roots, and apply the method to solve real-life mathematical problems involving dimensions, speed, and motion. With step-by-step explanations and board-focused examples, this topic strengthens your algebraic problem-solving skills.

2.0Learning Outcomes

After completing this topic, you will be able to:

Identify a quadratic equation in standard form.
Apply the quadratic formula to solve equations.
Understand the meaning of the discriminant.
Determine whether the roots are real, equal, or imaginary.
Solve application-based problems using the quadratic formula.
Compare factorization and quadratic formula methods.
Solve NCERT, competency-based, and CBSE Board examination questions confidently.

3.0Introduction to the Quadratic Formula

The Quadratic Formula is one of the most powerful methods for solving quadratic equations. While some equations can be solved easily by factorization, many cannot be factorized into simple integers. In such cases, the quadratic formula provides a direct and reliable method to find the roots of any quadratic equation.

In this topic, you'll learn how to derive and apply the quadratic formula, determine the nature of roots using the discriminant, and solve board-level numerical and application-based questions with confidence. This formula is an essential part of Class 10 Mathematics and frequently appears in CBSE Board examinations.

A quadratic equation, therefore, refers to a mathematical relationship between a variable and the constants, where the highest power of the variable is squared. Sometimes, solving quadratic equations can be difficult, especially when factoring does not work or is not done. One of the best methods available for their solution is the quadratic formula, the famous formula which gives the solutions directly.

4.0General Form of Quadratic Equation

The General Form of Quadratic Equation is an important aspect of solving quadratic equations as it allows systematic methods for finding the solutions of the equation. The standard form of quadratic equation can be written as:

$a x^{2} + b x + c = 0$

Here,

x is the variable or unknown we are solving for,
a, b, and c are constants known as the coefficients, and
$a \neq = 0$ (if a = 0, the equation would be linear, not quadratic, because the highest power of x would then be 1)

5.0Quadratic Equation Formula

The quadratic equation formula, also known as the quadratic formula, is used for finding the roots of the equation in a standard form, which is $(a x^{2} + b x + c = 0)$ . It provides solutions or roots of the equation without a need to use other tougher methods. The Quadratic Formula can be expressed as:

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Here, “ $(b^{2} - 4 a c)$ ” is known as the discriminant, which can be symbolised as “D”. It helps in determining the nature of roots:

If D＞0, then the roots of the equation will have two real “distinct” roots.
If D=0, then the equation will have two “equal” roots.
If D＜0, then the roots of the equation will be imaginary.

6.0How to Solve Quadratic Equations by Quadratic Formula

Solving quadratic equations can become easy by using the following four steps:

Step 1: Identify the values of a, b, and c from the quadratic equation.
Step 2: Put these values into the quadratic formula.
Step 3: Reduce the expression under the square root and determine whether the discriminant is positive, zero, or negative. The two former cases translate to two real solutions and one real solution, respectively. The latter gives complex solutions.
Step 4: Solve for x.

7.0Derivation of Quadratic Equation

The formula for the quadratic equation can be derived by using the standard form of the quadratic equation. Like this:

$a x^{2} + b x + c = 0$

Divide the equation by the “a”

$x^{2} + (\frac{b}{a}) x + (\frac{c}{a}) = 0$ …..(1)

Using the completing the square method, taking the coefficient of x, dividing it by ½, and then squaring it:

$\frac{b}{a} \Rightarrow \frac{b}{2 a} \Rightarrow \frac{b ^{2}}{4 a ^{2}}$

Adding and subtracting $\frac{b ^{2}}{4 a ^{2}}$ on both sides in equation (1)

$x^{2} + (\frac{b}{a}) x + \frac{b ^{2}}{4 a ^{2}} + (\frac{c}{a}) - \frac{b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} + \frac{c}{a} - \frac{b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} + \frac{4 a c - b ^{2}}{4 a ^{2}} = 0$

$(x + \frac{b}{2 a})^{2} = \frac{- ( 4 a c - b ^{2} )}{4 a ^{2}}$

$(x + \frac{b}{2 a}) = \pm \frac{b ^{2} - 4 a c}{4 a ^{2}}$

$x = - \frac{b}{2 a} \pm \frac{b ^{2} - 4 a c}{2 a}$

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

8.0Graphing Quadratic Equations

Graphing quadratic equations is a graphical method of finding the solutions, or roots, of a quadratic equation and understanding the behaviour of the equation written in the standard form $(a x^{2} + b x + c = 0)$ . This equation produces a parabola, which is a symmetrical curve. The shape of the parabola and its orientation (whether it opens upwards or downwards) depends on the value of a.

Steps for Graphing Quadratic Equations

Write the quadratic equation in its standard form.
Find the vertex of x - coordinate by using the formula $(x = \frac{- b}{2 a})$ and substituting the values of a and b. To find the y coordinate, simply put this value of the x-coordinate in the quadratic equation.
Find the axis of symmetry, which is a vertical passing through the vertex by using again the formula of $(x = \frac{- b}{2 a})$
Evaluate y when x = 0 and the y-intercept is (0, c) to find the y-intercept.
Draw the parabola by smoothly connecting the points, ensuring the curve opens upward (if a > 0) or downward (if a < 0).
The point where the parabola intercepts on the x-axis is the root of the quadratic equations.

9.0Quadratic Equation Examples

Problem 1: Find the roots of x²+ 6x − 7 = 0 using the quadratic formula.

Solution: Given that, x²+ 6x − 7 = 0

a = 1, b = 6, c = –7

$D = b^{2} - 4 a c = 6^{2} - 4 \times 1 \times (- 7)$

$D = 36 + 28 = 64$

$x = \frac{- 6 \pm 64}{2 ( 1 )} = \frac{- 6 \pm 8}{2}$

$x = \frac{- 6 + 8}{2}, x = \frac{- 6 - 8}{2}$

$x = 1, - 7$

Problem 2: Determine if the given equation x²+ 2x + 5 = 0 has a real root or not, if the roots are real, then find the roots of the equation.

Solution: Given that x²+ 2x + 5 = 0

a = 1, b = 2, c = 5

$D = b^{2} - 4 a c = 2^{2} - 4 (1 \times 5) = 4 - 20$

D = –16

Hence, the root of the given equation is imaginary.

Problem 3: The roots of the following equation x²− bx + 9 = 0 are equal, which is 3. Find the value of b.

Solution: Given that x²− bx + 9 = 0

a = 1, c = 9, b = ?

As the roots of the equations are equal hence, D = 0

$D = b^{2} - 4 a c$

$0 = b^{2} - 4 \times 1 \times 9$

$b^{2} = 36$

$b = 36 = 6$

10.0EUREKA by ALLEN – The Future of Class 10 Learning

EUREKA by ALLEN is an advanced online learning platform designed for Class 10 students preparing for CBSE and State Board exams. With interactive story-based lessons, expert faculty guidance, AI-powered doubt-solving, and board-focused practice, EUREKA helps students strengthen concepts, improve retention, and score higher in class 10 board examinations.

Key Features of EUREKA Class 10 Courses:
Easy-to-follow concept explanations Story-based learning for better retention Exam-oriented preparation strategy Answer writing practice sessions Instant quizzes and self-assessment Performance tracking dashboard Round-the-clock AI doubt support NCERT-based curriculum coverage Study at your own pace

Explore Now

11.0Supporting Study Materials

These CBSE notes and NCERT solutions for the chapter “Polynomials”, covering topics like Quadratic Formulas, are prepared in accordance with the latest CBSE Class 10 Maths syllabus and NCERT guidelines. The study material offers detailed explanations of important concepts, definitions, examples, and exam-orientated questions to help students understand redox reactions, corrosion, and rancidity effectively while strengthening their preparation for board examinations.

CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations	NCERT Solutions for Class 10 Maths Chapter 4: Quadratic Equations

12.0Previous Year Questions (PYQs) on Quadratic Formula

Question: A quadratic equation has a = 1, b = –8, and c = 16. Determine the nature of its roots.

Answer

Use the discriminant:

Δ = (–8)² – 4(1)(16)

= 64 – 64

= 0

Therefore, The equation has two equal real roots.

Question: Find two consecutive integers whose product is 210.

Answer: Let the integers be x and x + 1.

x(x + 1) = 210

x² + x – 210 = 0

Using the quadratic formula,

x = 14 Therefore, The integers are 14 and 15.

30-Second Quick Review: Quadratic Formula

Standard form: ax² + bx + c = 0, where a ≠ 0
The Quadratic Formula can solve every quadratic equation.
Discriminant: Δ = b² – 4ac
If Δ > 0 → Two distinct real roots.
If Δ = 0 → Two equal real roots.
If Δ < 0 → No real roots.
The ± sign gives two possible solutions.
The quadratic formula is especially useful when factorization is difficult or impossible.

Frequently Asked Questions

Can a quadratic equation have more than two solutions?

What happens when a quadratic equation has a negative discriminant?

Can I always factor a quadratic equation?

What are the roots or solutions of a quadratic equation?

Frequently Asked Questions

Can a quadratic equation have more than two solutions?

What happens when a quadratic equation has a negative discriminant?

Can I always factor a quadratic equation?

What are the roots or solutions of a quadratic equation?

Join ALLEN!

Quadratic Formula

1.0Master the Quadratic Formula in Minutes

2.0Learning Outcomes

3.0Introduction to the Quadratic Formula

4.0General Form of Quadratic Equation

5.0Quadratic Equation Formula

6.0How to Solve Quadratic Equations by Quadratic Formula

7.0Derivation of Quadratic Equation

8.0Graphing Quadratic Equations

9.0Quadratic Equation Examples

10.0EUREKA by ALLEN – The Future of Class 10 Learning

11.0Supporting Study Materials

12.0Previous Year Questions (PYQs) on Quadratic Formula

30-Second Quick Review: Quadratic Formula

Recommended Next Topics

Table of Contents

Join ALLEN!

Quadratic Formula

1.0Master the Quadratic Formula in Minutes

2.0Learning Outcomes

3.0Introduction to the Quadratic Formula

4.0General Form of Quadratic Equation

5.0Quadratic Equation Formula

6.0How to Solve Quadratic Equations by Quadratic Formula

7.0Derivation of Quadratic Equation

8.0Graphing Quadratic Equations

9.0Quadratic Equation Examples

10.0EUREKA by ALLEN – The Future of Class 10 Learning

11.0Supporting Study Materials

12.0Previous Year Questions (PYQs) on Quadratic Formula

30-Second Quick Review: Quadratic Formula

Recommended Next Topics

Table of Contents