Lesson 1: Solving Quadratic Equations

Property

Quadratic equations are equations of the form $ax^2 + bx + c = 0$, where $a \neq 0$.
They differ from linear equations by including a term with the variable raised to the second power.
We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.

Examples

• The equation $x^2 = 100$ is a quadratic equation where the $x$ is squared.
• An equation like $5y^2 + 10y - 15 = 0$ is a quadratic equation in standard form.
• The equation $2z^2 = 50$ is a quadratic equation of the form $ax^2 = k$.

Explanation

Think of a quadratic equation as a 'squared' equation. Unlike simple linear equations, you can't isolate the variable with basic arithmetic. It requires special tools like factoring or using square roots to find the solution(s).

Property

To Solve a Quadratic Equation by Graphing:

1. Graph the quadratic function $y = ax^2 + bx + c$
2. Find where the parabola crosses the $x$-axis (the $x$-intercepts)
3. The $x$-coordinates of these intersection points are the solutions to the equation $ax^2 + bx + c = 0$

Property

To solve a quadratic equation using the Square Root Property:

1. Isolate the quadratic term and make its coefficient one.
2. Use the Square Root Property.
3. Simplify the radical.
4. Check the solutions.

Examples

• To solve $4x^2 = 100$, first divide by 4 to get $x^2 = 25$. Then, using the Square Root Property, $x = \pm\sqrt{25}$, so $x = 5$ and $x = -5$.
• Solve $3y^2 - 54 = 0$. First, add 54 to both sides to get $3y^2 = 54$. Divide by 3 to get $y^2 = 18$. The solution is $y = \pm\sqrt{18} = \pm 3\sqrt{2}$.
• Solve $3c^2 = 75$. Divide by 3 to get $c^2=25$. Using the property, $c = \pm\sqrt{25}$, so $c=5, c=-5$.

Explanation

Before you can apply the Square Root Property, the $x^2$ term must be by itself. To achieve this, divide both sides of the equation by the coefficient of $x^2$, and then proceed with taking the square root.