Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

9.1.3 Solving Multi Step Quadratic Equations Using Perfect Squares

9.1.3 • Solving Multi Step Quadratic Equations Using Perfect Squares

Activity

Han and Jada solved the same quadratic equation with different methods. Here they are:

Han's Method	Jada's Method
$\begin{array}{rcl} (x - 6)^{2} & = & 25 \\ (x - 6) (x - 6) & = & 25 \\ x^{2} - 12 x + 36 & = & 25 \\ x^{2} - 12 x + 11 & = & 0 \\ (x - 11) (x - 1) & = & 0 \\ (x - 11) & = & 0 \\ (x - 1) & = & 0 \\ \begin{array}{rcl} x & = & 11 \end{array} & o r & x = 1 \end{array}$	$\begin{array}{rcl} (x - 6)^{2} & = & 25 \\ \sqrt{(x - 6)^{2}} & = & \sqrt{25} \\ x - 6 & = & \pm 5 \\ x - 6 & = & 5 \\ x - 6 & = & - 5 \\ x = 11 & o r & x = 1 \end{array}$

Work with a partner to solve these equations. For each equation, one partner solves with Han's method, and the other partner solves with Jada's method. Make sure both partners get the same solutions to the same equation. If not, work together to find your mistakes.

1.

$(y - 5)^{2} = 49$

Compare your answer:

$y = 12$ and $y = - 2$

2.

$(x + 4)^{2} = 9$

Compare your answer:

$x = - 1$ and $x = - 7$

3.

$(z + \frac{1}{3})^{2} = 49$

Compare your answer:

$z = 6 . \overset{―}{6}$ or $6 \frac{2}{3}$ and $z = - 7 . \overset{―}{3}$ or $- 7 \frac{1}{3}$

4.

$(v - 0.1)^{2} = 0.36$

Compare your answer:

$v = 0.7$ and $v = - 0.5$

Video: Learning About Solving Multi Step Quadratic Equations Using Perfect Squares

Watch the following video to learn more about solving multi step quadratic equations using perfect squares.

Learning About Solving Multi Step Quadratic Equations Using Perfect Squares

Self Check

Find all solutions to

(x-3)^2=25

.

$x=28$
$x=22$ , $x=28$
$x=-2$ , $x=8$
$x=8$

Additional Resources

Two Ways to Solve Quadratic Equations

In the activity, Han and Jada used two different methods for solving a quadratic equation, using the distributive property and the square root property.

Using the Distributive Property to Solve Quadratic Equations

How to Solve a Quadratic Equation Using the Distributive Property:

Step 1 - Isolate the quadratic term and make its coefficient 1.

Step 2 - Rewrite the term so the binomial is multiplied by itself.

Step 3 - Use the distributive property or FOIL.

Step 4 - Combine like terms so that the equation is set equal to 0.

Step 5 - Factor.

Step 6 - Set each factor equal to 0.

Step 7 - Solve.

Step 8 - Check the solutions.

Example

Solve $4 (x - 7)^{2} = 36$

Step 1 - Isolate the quadratic term (divide by 4).
$(x - 7)^{2} = 9$

Step 2 - Rewrite the term so the binomial is multiplied by itself.

$(x - 7) (x - 7) = 9$

Step 3 - Use the distributive property or FOIL.

$x^{2} - 7 x - 7 x + 49 = 9$

Step 4 - Combine like terms so that the equation is set equal to 0.

$x^{2} - 14 x + 40 = 0$

Step 5 - Factor.

$(x - 10) (x - 4) = 0$

Step 6 - Set each factor equal to 0.

$x - 10 = 0$ , $x - 4 = 0$

Step 7 - Solve.

$x - 10 = 0 x - 4 = 0$

$x = 10 x = 4$

Step 8 - Check the solutions.

$4 (10 - 7)^{2} = 36 4 (4 - 7)^{2} = 36$

$4 (3)^{2} = 36 4 (- 3)^{2} = 36$

$36 = 36 36 = 36$

Try it

Using the Distributive Property to Solve Quadratic Equations

Use the distributive property to solve $3 (x - 4)^{2} = 48$ .

Here is how to solve the quadratic equation using the distributive property.

Step 1 - Isolate the quadratic term (divide by 3).

$(x - 4)^{2} = 16$

Step 2 - Rewrite the term so the binomial is multiplied by itself.

$(x - 4) (x - 4) = 16$

Step 3 - Use the distributive property or FOIL.

$x^{2} - 4 x - 4 x + 16 = 16$

Step 4 - Combine like terms so that the equation is set equal to 0.

$x^{2} - 8 x = 0$

Step 5 - Factor.

$x (x - 8) = 0$

Step 6 - Set each factor equal to 0.

$x = 0$ or

$x - 8 = 0$

Step 7 - Solve.

$x = 0 x - 8 = 0$

$x = 0 x = 8$

Step 8 - Check the solutions.

$3 (0 - 4)^{2} = 48 3 (8 - 4)^{2} = 48$

$3 (- 4)^{2} = 48 3 (4)^{2} = 48$

$48 = 48 48 = 48$

If $x^{2} = k$ , then

$x = \sqrt{k}$ or $x = - \sqrt{k}$ or $x = \pm \sqrt{k}$

How to Solve a Quadratic Equation Using the Square Root Property:

Step 1 - Isolate the quadratic term and make its coefficient one.

Step 2 - Use the square root property. Take the square root of both sides of the equation and remember the "plus or minus."

Step 3 - Rewrite the equation as two equations.

Step 4 - Solve each equation.

Step 5 - Check the solutions.

Example

Solve $4 (x - 7)^{2} = 36$

Step 1 - Isolate the quadratic term (divide by 4).

$(x - 7)^{2} = 9$

Step 2 - Use the square root property.

$\sqrt{(x - 7)^{2}} = \sqrt{9}$

$x - 7 = \pm 3$

Step 3 - Rewrite the equation as two equations.

$x - 7 = - 3$

$x - 7 = 3$

Step 4 - Solve each equation.

$\begin{matrix} x - 7 = - 3 & x - 7 = 3 \\ x = 4 & x = 10 \end{matrix}$

Step 5 - Check the solutions.

$\begin{array}{l} 4 (4 - 7)^{2} = 36 & 4 (10 - 7)^{2} = 36 \\ 4 (- 3)^{2} = 36 & 4 (3)^{2} = 36 \\ 36 = 36 & 36 = 36 \end{array}$

Try it

Using the Square Root Property to Solve Quadratic Equations

Use the square root property to solve $3 (x - 4)^{2} = 48$ .

Here is how to solve the quadratic equation using the square root property:

Step 1 - Isolate the quadratic term (divide by 3).

$(x - 4)^{2} = 16$

Step 2 - Use the square root property.

$\sqrt{(x - 4)^{2}} = \sqrt{16}$

$x - 4 = \pm 4$

Step 3 - Rewrite the equation as two equations.

$x - 4 = 4 x - 4 = - 4$

Step 4 - Solve each equation.

$x - 4 = 4 x - 4 = - 4$

$x = 8 x = 0$

Step 5 - Check the solutions.

$3 (8 - 4)^{2} = 48 3 (0 - 4)^{2} = 48$

$3 (4)^{2} = 48 3 (- 4)^{2} = 48$

$48 = 48 48 = 48$