What is a Quadratic Equation?
A quadratic equation is an equation that can be written as:
$$ ax^2 + bx + c $$ where a ≠ 0.
A quadratic equation must have a squared term as its highest power.
Examples of quadratic equations
$$ y = 5x^2 + 2x + 5 \\ y = 11x^2 + 22 \\ y = x^2 - 4x +5 \\ y = -x^2 + 5 $$
Non Examples
$$ y = 11x + 22 \\ y = x^3 -x^2 +5x +5 \\ y = 2x^3 -4x^2 \\ y = -x^4 + 5 $$
What is a the solution of a Quadratic Equation?
The solution of a quadratic equation is the value of x when you set the equation equal to $$ \red {\text {zero}}$$
i.e. When you solve the following general equation: $$\red 0 = ax^2 + bx + c $$.
There are many ways to solve quadratic equations. One of the ways is to factor the equation.
General Steps to solve by factoring
Create a factor chart for all factor pairs of c.
A factor pair is just two numbers that multiply and give you c.
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Out of all of the factor pairs from step 1, look for the pair (if it exists) that add up to b
Note: if the pair does not exist, you must either complete the square or use the quadratic formula.
- Insert the pair you found in step 2 into two binomals.
Solve each binomial for zero to get the solutions of the quadratic equation.
Example of how to solve a quadratic equation by factoring
Quadratic Equation: y = x² + 2x + 1.
Below is a picture representing the graph of y = x² + 2x + 1 as well as the solution we found by factoring.
Practice 1
Below is a picture of the graph of the quadratic $$ y = x^2 - 2x + 1 $$as well as the solutions.
Practice 2
Below is a picture of the graph of the quadratic $$ y = x^2 + 4x + 4 $$ as well as the solutions.
Practice 3
$$ y = x^2 - 4x + 4 $$ is graphed below as well as its solution (2, 0).
Practice 4
$$ y = x^2 + 6x + 9 $$ is graphed below as well as its solution (2, 0).
Practice 5
y = x² − 6x + 9
- c = 9
- b = 6
- The only factors of c whose sum is b are -3 • -3.
- y = (x − 3)(x − 3)
- 0 = (x − 3)
- The solution is at x = 3.
Practice 6
$$ y = x^2 + 2x - 3$$ is graphed below as well as both the solutions.
Practice 7
$$ y = x^2 - 2x - 3 $$ is graphed below as well as both the solutions.
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