**What is a quadratic equation? **A polynomial in which the highest exponent is 2 is called a quadratic equation The standard form of a quadratic expression is ax2 + bx + c. Equate the quadratic expression to 0 and you get the standard form of the quadratic equation: ax2 + bx + c = 0 in the quadratic equation, a is the leading coefficient. a and b are called coefficients of x2 and x, while c is called a constant term.

**What are not quadratic expressions?**

### Example 1:

√x2 – 5x + 6 is not a quadratic equation, as the highest exponent of x is not 2 for √x2 is same as x, in which the exponent is 1 but not 2

### Example 2:

*(x – 2) (x – 3) = (x + 1) (x – 4)* is also not a quadratic equation as the highest exponent of x on simplifying is not 2, but 1.

### Example 3:

*x2 + 1/ x2 = 0* is not a quadratic equation, as the highest exponent of the polynomial is 4 and not 2.

### Example 4:

*x2 + 2√x + 1 = 0* is not a quadratic equation, as the power of x, in the second term 2√x, is 1/2, not an integer. (See definition of a Polynomial)

## Roots of a quadratic equation

Roots of a quadratic equation are values of the variable for which the quadratic expression becomes equal to 0.

### Example 1:

Consider the quadratic equation:* x2 – 2x + 1 = 0.*

If x = 1, then the quadratic expression *x2 – 2x + 1* becomes: *12 – 2(1) + 1 = 0*.

This value 1 of the variable x, for which the quadratic equation reduces to 0 is called the root of the quadratic equation: *x2 – 2x + 1 = 0*.

### Example 2:

Consider the quadratic equation: *x2 – 5x + 6 =* 0. Plug 2 or 3 in x and the quadratic expression *x2 – 5x + 6* becomes equal to 0. So, 2 or 3 are roots of the quadratic equation *x2 – 5x + 6 = 0.*

**Note:** Any quadratic equation can have at most two roots, i.e. one or two roots, but not more than 2.

- In example 1 above, x = 1 is the only root. When both the roots of a quadratic equation are equal, we call the root a “double root”. In example 1 above, 1 is a double root
- In example
*2 above, x = 2 or x = 3*are the two roots. In this quadratic equation, the two roots are real and different. - How to solve a quadratic equation: To solve a quadratic equation is to find its roots. We will discuss four methods of solving a quadratic equation, i.e. finding roots. They are: a factorization method, substitute and factorize, by completing the square, quadratic formula method.

**Let us capture a brief overview of each method:**

## 1. Factorization method

### Example 1:

Solve the quadratic *x2 – 3x + 2 = 0*

Solution: You must be thorough with factoring methods learnt in factoring lesson.

Factorize:

*x2 – 2x – x + 2 = 0,*

*x(x – 2) – 1(x – 2) = 0,*

*(x – 2) (x – 1) = 0,*

*So, x – 2 = 0 or x – 1 = 0,*

*x = 2 or x = 1* are the two roots of the given quadratic equation. in short, 2 or 1 are the roots

**Note:** Use “or” to connect the two roots. Do not use “and” to connect the two roots. Do not say 2 “and” 1 are roots.

## 2. Substitute and Factorize:

### Example 1:

Solve the quadratic equation: *51 + x + 5 1 – x = 26*

Solution: first simplify the quadratic as:

*5 × 5x + 5/5x = 26*

*Substitute y for 5x.*

*Now the quadratic is: 5y + 5/y = 26,*

*5y2 + 5 = 26y, i.e.*

*5y2 – 26y + 5 = 0,*

*5y2 – 25y –y + 5 = 0,*

*5y(y – 5) – 1(y – 5) = 0,*

*(5y – 1) (y – 5) = 0,*

*5y – 1 = 0 or y – 5 = 0,*

*y = 1/5 or y = 5,*

*Now put 5x = 1/5 or 5x = 5,*

*5x = 5 – 1 or 5x = 5,*

*x = – 1 or x = 1.*

## 3. Completing the square method:

### Example 1:

*x2 + 10x = 75*

Solution: add 25 to each side to complete the square on the left side:

*x2 + 10x + 25 = 75 + 25*

*x2 + 10x + 25 = 100,*

*(x + 5)2 = 102,*

*x + 5 = 10 or x + 5 = -10*

*x = 5 or x = -15*

## 4. Using the quadratic formula:

For the quadratic equation *ax2 + bx + c = 0*, the two roots are:

*x = [–b – √ (b2 – 4ac)]/2a,*

*x = [–b + √ (b2 – 4ac)]/2a*

### Example 1:

Find the roots of the quadratic equation *x2 – 9x + 36 = 0* using quadratic formula.

Solution:

*In the quadratic x2 – 13x + 36 = 0:*

*a = 1, b = – (-13) = 13 and c = 36,*

*b2 – 4ac = 132 – 4 × 1 × 36 = 169 – 144 = 25*

so, of the two roots, one root is :

*x = [–b–√ (b2 – 4ac)]/2a,*

*x = [13 – √ (25)]/2 ×1 = [13 – 5 ]/2 = 8/2 = 4*

and the other root is:

*x = [–b + √ (b2 – 4ac)]/2a*

*x = [13 + √ (25)]/2 ×1 = [13 + 5]/2 = 18/2 = 9*

## 5. Nature of roots of a quadratic equation using the discriminant

By nature of roots is meant whether roots are real or complex and equal or different. b2 – 4ac is called discriminant of a quadratic equation. We use the discriminant to find the nature of roots of a quadratic equation.

## 6. Sum and Product of roots of a quadratic equation:

In the quadratic equation ax2 + bx + c = 0

- Sum of the roots = – (b/2) = – (coefficient of x)/ (coefficient of x2)
**Example1:**In the quadratic equation:*3×2 – 12x + 36 = 0,*sum of the roots is*– (-12)/3 = 4*- Product of roots = c/a = (constant term) / (leading coefficient) [Leading coefficient is same as coefficient of x2]

### Example 2:

In the quadratic equation: *3×2 – 12x + 36 = 0*, product of the roots is *36/3 = 12*

### Example 1:

In the quadratic equation *9×2 – 12x + 36 = 0*, sum of the roots is *– (- 12/9) = 12/9 = 4/3* and product of roots is *36/12 = 3.*

## 7. Signs of roots of a quadratic equation:

Consider the quadratic equation *ax2 + bx + c = 0*