Page 110 - Algebra

6. Quadratic Equation
A quadratic equation in x is always of the form ax2 + bx + c = 0, where a≠0 and a, b, and c are real numbers. In other words, a polynomial of degree 2 is called a quadratic equation.
For example,
x2 – 3x + 6 = 0 is a quadratic equation
Roots of a quadratic equation
Let’s understand it with an example:
If you want to find the factors of a quadratic equation ax2 + bx + c = 0 This is not of the form of any identity you have previously studied. So, we will split the middle term as mx + nx.
The value of m and n should be such that m + n = b and mn = c
Worked Example
Factorize the following quadratic equation:
4x2 + 8x + 3 = 0
Solution:
4x2 + 8x + 3 = 0
Short cut to find the roots is multiply ‘a’ and ‘c’ in the equation ax2 + bx + c = 0 Here a = 4, c = 3
ac = 12
Now write 12 as a sum or difference of two numbers such that it is equal to 8 (which is ‘b’ in the quadratic equation), and the product is equal to 12.
You can write 12 as
12 = 2 × 6 = 3 × 4 = 1 × 12
Out of these factors the sum of 6 and 2 results in 8.
4x2 + 2x + 6x + 3 =0 2x(2x + 1) + 3(2x + 1) = 0
(2x + 3) (2x + 1) = 0
13
x = – and x = – 22
Page 109 of 177
Algebra I & II