Roots of Quadratic Equation

A quadratic equation is a form of polynomial equation that plays an important role in algebra and mathematics in general. It is termed "quadratic" because the highest power of the variable is 2, making it a second-degree equation. In simple words, a quadratic equation refers to a mathematical relationship between a variable and the constants, where the highest power of the variable is squared. 

1.0General Form Quadratic Equation

To solve the quadratic equations systematically and easily, firstly, it must be converted into its standard form also known as the general form of the quadratic equation. The equations are written in general form as: 

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

Here, 

• a, b, and c are the constants of the equation known as the coefficients, with a 0. 
• x is the variable with x² 0 to ensure the basic form of the quadratic equations. 

2.0Determinant of Quadratic Equations

The discriminant, often referred to as the determinant, is an important element in understanding the nature of the roots of a quadratic equation. The determinant of a quadratic equation, $a{x}^2+bx+c=0$, can be written as: 

$D=b^2-4ac$

The nature of quadratic means whether the root of the equation is real or complex/imaginary. This is how discriminant decides the nature of the roots: 

1. If D＞0, the roots of the given equation will have two real “distinct” roots.
2. If D=0, the given equation will have two “equal” roots. 
3. If D＜0, the roots of the given equation will be imaginary or complex.

3.0Types of Roots of a Quadratic Equation

 As mentioned earlier, the Roots of a quadratic equation can be classified based on the determinant of the algebraic equation, which includes: 

1. Real Roots of Quadratic Equation: 

Real roots are the solutions to the quadratic equation, which are real numbers. These roots occur only when the discriminant D ≥ 0. The real roots may be distinct or equal:

• Distinct Real Roots: When the discriminant D > 0, the quadratic equation has 2 distinct real roots.
• Equal Real Roots: If the discriminant D = 0, then the quadratic equation has 2 equal real roots, which is also known as a double root.

2. Complex Roots of Quadratic Equation: 

The Complex or imaginary roots of a quadratic equation is the case where the discriminant D < 0, indicating that the given quadratic equation has no real roots but rather complex roots. These roots are conjugate of each other and the imaginary part of the equation is written as “i” where $i=\sqrt{-1}.$

The expression for complex roots is given in the general form:

$x_1=\frac{-b+i\sqrt{-D}}{2a}$  and  $x_2=\frac{-b-\sqrt{-D}}{2a}$

4.0Calculating the Quadratic Equations 

Solving quadratic equations to find the roots of the equation can become easy by using the most commonly used three methods of finding roots of the algebraic equations. Which includes: 

1. Factorisation Method: 

The factorisation method includes rewriting quadratic equations in factored form and then finding the value of x. The steps to solve the equation involve the following: 

• Multiply “a” the coefficient of x² with the constant term of c, resulting in the new term ac. 
• Now split the middle term (coefficient of x) of the standard form into two numbers such that it gives the number ac after multiplying and adding both terms. 
• Group the first and last two terms and factor each group by finding the common term in them. 
• Now solve for x.

2. Completing the Square Method: 

The method involves transforming the quadratic equation into a perfect square trinomial, which can then be solved by taking the square root. The steps to solve the equation are: 

• Write the quadratic equation in standard form $a{x}^2+bx+c=0$
• Make sure a = 1, if not, divide the whole equation with a. 
• Now take b and then follow this sequence of operations 
• • b $\Rightarrow$half of b that is b/2 $\Rightarrow$squaring b/2 which now will be $b^2/4$
• Now, add and subtract $\frac{b^2}{4}$ simultaneously in the equations. This will create a perfect square trinomial. 
• Solve for x. 

3. Quadratic Formula: 

Complex quadratic which are difficult to factorise, can directly be solved with the help of a formula for roots of quadratic equations, which can expressed as: 

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

5.0Solved Examples

Problem 1: Find the roots of a quadratic equation x² + 5x + 6 = 0, using the factorisation method. 

Solution: Given that x² + 5x + 6 = 0

x² + 3x + 2x + 6 = 0

x(x + 3) + 2(x + 3) = 0

(x + 2)(x + 3) = 0 

x = –2, –3 

Problem 2: calculate roots of quadratic equation x² + 8x − 9 = 0, using completing the square method. 

Solution: Given that x² + 8x − 9 = 0

Follow these operations b $\Rightarrow$half of b that is b/2$\Rightarrow$squaring b/2 which now will be $b^2/4$

8 $\Rightarrow$8/2=4 $\Rightarrow$$4^2$= 16

Adding and subtracting 16 in the equation 

x² + 8x + 16 − 9 – 16 = 0

(x + 4)² – 25 = 0

(x + 4)² – 5² = 0 

(x + 4 + 5)(x + 4 – 5) = 0

x = 9, –1

Problem 3: find the roots of the equation x²−4x+3=0, using the quadratic formula. 

Solution: Given a = 1, and b = –4, c = 3 

$D=b^2-4ac$

$D=(-4{)}^2-4\times 1\times 3$

$D=16-12=4$

Using the quadratic formula: 

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

$x=\frac{-(-4)\pm \sqrt{4}}{2(1)}$

$x=\frac{4\pm 2}{2}$

$x=\frac{4+2}{2},\frac{4-2}{2}$

$x=3,1$