Page 112 - Algebra

6.1. Factoring & Finding Roots of a Quadratic Equation
• Factorization is the process of writing an expression as a product of two or more expressions. For example,
2x3 + 4x2 – 8x can be factorized as 2x(x2 + 2x – 4)
• You can factorize a binominal by taking out the greatest common factor from the expression. In the above example, the GCF is 2x.
6.2. Nature of Roots
The roots of a quadratic equation ax2 + bx + c = 0 is given by
−𝑏±√𝑏2−4𝑎𝑐 2𝑎
x=
When b2 – 4ac > 0, you get two distinct roots –𝑏+√𝑏2−4𝑎𝑐 and –𝑏−√𝑏2−4𝑎𝑐
2𝑏 When b – 4ac = 0, you get two roots and both are equal. The roots are – . 2𝑎
When b2 – 4ac < 0, you get no real roots for the quadratic equation.
As you can see, b2 – 4ac determines the nature of the roots of a quadratic equation, it’s called discriminant.
2𝑎 2𝑎
REMEMBER:
A quadratic equation has:
• Two distinct roots when b2 – 4ac > 0
• Twoequalrootswhen b2 –4ac =0
• No real roots when b2 – 4ac < 0
Worked Example
Find the nature of the roots of the following quadratic equation:
2x2 – 3x + 5 = 0
Solution:
a =2
b= – 3 c=5
b2 –4ac=(
= 9 – 40 = – 36
– 3)2 – 4(2)(5)
So, the quadratic equation has no real roots.
Page 111 of 177
Algebra I & II