Lesson 4: Solving Quadratic Equations Using Square Roots

Property

Taking a square root is the opposite of squaring a number.
To solve an equation of the form $x^2 = k$ (where $k > 0$), we take the square root of both sides.
Because a positive number has two square roots, the solution is written as:

Examples

• To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = -9$.

Property

To solve a quadratic equation using the square root property:
Step 1. Isolate the quadratic term and make its coefficient one. For an equation like $ax^2 = k$, this means rewriting it as $x^2 = \frac{k}{a}$.
Step 2. Use the Square Root Property: $x = \pm\sqrt{\frac{k}{a}}$.
Step 3. Simplify the radical.
Step 4. Check the solutions.

Examples

• Solve $5x^2 = 125$. First, divide by 5 to isolate $x^2$, giving $x^2 = 25$. Then, use the Square Root Property to find $x = \pm\sqrt{25}$, so the solutions are $x=5$ and $x=-5$.
• Solve $\frac{3}{4}u^2 + 2 = 29$. Subtract 2 to get $\frac{3}{4}u^2 = 27$. Multiply by $\frac{4}{3}$ to get $u^2 = 36$. The solutions are $u = \pm\sqrt{36}$, so $u=6$ and $u=-6$.
• Solve $3x^2 - 9 = 54$. Add 9 to get $3x^2 = 63$. Divide by 3 to get $x^2 = 21$. The solutions are $x = \pm\sqrt{21}$.

Explanation

Before you can use the Square Root Property, the $x^2$ term must be by itself. This method is about clearing away any coefficients or constants first. Once you have $x^2$ isolated, you can solve for the two possible values of $x$.

Property

To solve equations of the form $ax² + b = c$, follow these steps:

1. Subtract $b$ from both sides: $ax² = c - b$
2. Divide both sides by $a$: $x² = \frac{c - b}{a}$
3. Take the square root of both sides: $x = ±\sqrt{\frac{c - b}{a}}$ (when $\frac{c - b}{a} \geq 0$)

Examples