1. This document discusses solving quadratic equations by factoring and using the quadratic formula.
2. To solve by factoring, rewrite the equation so one side equals 0, factor the non-zero side, and set each factor equal to 0 to solve.
3. The quadratic formula is x = (-b ± √(b^2 - 4ac))/2a. This formula can be used to solve any quadratic equation in the form ax^2 + bx + c = 0.
2. Solution of a quadratic equation
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An equation is a quadratic equation if it can be written in the form:
𝑎𝑥2
+ 𝑏𝑥 + 𝑐 = 0
where a, b, c are known numbers, a ≠ 0
Examples
▪ 4𝑤2 − 10 = 0 is a quadratic equation (it is in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0
where 𝑏 = 0 and instead of x as our unknown variable we have 𝑤).
▪ 3(2𝑥2
+ 1) = −5𝑥 is a quadratic equation (we can first expand the left-
hand side into 6𝑥2
+ 3 = −5𝑥, then rearrange it into 6𝑥2
+ 5𝑥 + 3 = 0).
▪ 7𝑥 − 4 is not a quadratic equation (because 𝑎 = 0).
3. Solve by Factoring
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To solve quadratic equations using factoring, follow these steps:
1. Rewrite/rearrange the quadratic equation so that one side is equal to 0
(This is very important for this method to work!).
2. Factor the other side of the equation (i.e. the non-zero side).
3. Set each of the factors equal to 0 and solve these linear equations
separately for roots.
6. Solve using the Quadratic Formula
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The quadratic formula:
𝑥 =
−𝑏 ± 𝑏2 −4𝑎𝑐
2𝑎
For this formula to work properly, you first need to make sure
that your quadratic equation is in the form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 .
We have to consider two expressions:
𝑥 =
−𝑏+ 𝑏2−4𝑎𝑐
2𝑎
and 𝑥 =
−𝑏− 𝑏2−4𝑎𝑐
2𝑎
7. Solve using the Quadratic Formula
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Example:
One number is the square of another number. If their sum is 176, what are
the two numbers? Round answers to two decimal places.
8. Solve using the Quadratic Formula
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To determine its solutions, we need to make one side equal to 0, then
factor it:
10. Number of roots
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To determine the number of roots a quadratic equation has, we can
use a part of the quadratic formula called the discriminant:
𝐷 = 𝑏2 − 4𝑎𝑐
There are three possible cases:
- If 𝑏2 − 4𝑎𝑐 < 0, then 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has no real roots.
- If 𝑏2 − 4𝑎𝑐 = 0 , then 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has one root.
- If 𝑏2 − 4𝑎𝑐 > 0 , then 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 has two (distinct) roots.
