Section 1.4
Quadratic Equations
Definition:
An equation that can be written in the form
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
where a, b, and c are real numbers with a ≠ 0, is quadratic equation. The given form is
called standard form.
1. Solving by Factoring:
To solve a quadratic equation by factoring, rewrite the equation, if necessary, so that one
side is equal to 0 and use the Zero-Product Property.
Zero-Product Property:
𝑎𝑏 = 0 if and only if 𝑎 = 0 or 𝑏 = 0.
Example 1: Solve the following equations by factoring.
a) 𝑥 2 − 3𝑥 = −2
b) 𝑥 2 + 3𝑥 = 18
c) 4𝑥 2 − 4𝑥 = −1
Section 1.4 Quadratic Equations
d) 6𝑥 2 + 7𝑥 = 3
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Square root Property:
If 𝑥 2 = 𝑘, then 𝑥 = √𝑘 or 𝑥 = −√𝑘
Example 2: Solve each quadratic equation.
a) 𝑥 2 = 17
c) (𝑥 − 4)2 = 1
b) 49𝑥 2 + 36 = 0
d) (2𝑥 − 1)2 = 12
2. Solving by Completing the Square:
To solve 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 with a ≠ 0, by completing the square, use these steps.
Step 1 If a ≠ 1, divide both sides of the equation by a.
Step 2 Rewrite the equation so that the constant term is alone on one side of the equality
symbol.
Step 3 Take a half of the coefficient of 𝑥, and then square the result. Now add this square
to each side of the equation.
Step 4 Factor the resulting trinomial as a perfect square and combine like terms on the
other side.
Step 5 Use the square root property to complete the solution.
Example 3: Solve 𝑥 2 − 4𝑥 − 14 = 0 by completing the square.
Step 1
Step 2
Step 3
Step 4
Step 5
Section 1.4 Quadratic Equations
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Example 4: Solve 9𝑥 2 − 12𝑥 + 9 = 0 by completing the square.
3. Solving by the Quadratic Formula:
The solution of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 with a ≠ 0 are given by the
quadratic formula.
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
Example 5: Find all solutions of the following each quadratic equation:
a) 𝑥 2 − 4𝑥 = −2
b) 2𝑥 2 = 𝑥 − 4
Example 6: Solve for the specified variable. Use ± when taking square roots.
Section 1.4 Quadratic Equations
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a) A =
𝜋𝑑2
4
, for 𝑑
b) 𝑟𝑡 2 − 𝑠𝑡 = 𝑘 (𝑟 ≠ 0), for 𝑡
The quantity under the radical in the quadratic formula, 𝑏 2 − 4𝑎𝑐, is called the
discriminant.
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
Example 7: Determine the number of distinct solutions, and tell whether they are rational,
irrational, or nonreal complex numbers.
a) 5𝑥 2 + 2𝑥 − 4 = 0
b) 𝑥 2 − 10𝑥 = −25
c) 2𝑥 2 − 𝑥 + 1 = 0
Section 1.4 Quadratic Equations
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