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