Lesson 3: Solving Quadratic Equations Using Square Roots

Property

If $x^2 = k$, and $k \ge 0$, then $x = \sqrt{k}$ or $x = -\sqrt{k}$. The solution can also be written as $x = \pm \sqrt{k}$.

Examples

• To solve $x^2 = 81$, we apply the Square Root Property to get $x = \pm\sqrt{81}$, which gives the solutions $x = 9$ and $x = -9$.
• For the equation $y^2 = 15$, the solutions are $y = \pm\sqrt{15}$, which we leave in radical form as $y = \sqrt{15}$ and $y = -\sqrt{15}$.
• If $z^2 = -9$, there is no real solution because the square root of a negative number is not a real number.

Explanation

When a variable squared equals a number, the variable itself can be either the positive or negative square root of that number. This is because squaring a negative value results in a positive value, creating two possible answers.

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:

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.