10.1 Solve Quadratic Equations Using the Square Root Property

Learning Objectives

By the end of this section, you will be able to:

  • Solve quadratic equations of the form ax^2=kax^2=k using the Square Root Property
  • Solve quadratic equations of the form a(x−h)^2=ka(x−h)^2=k using the Square Root Property

BE PREPARED 10.1

Before you get started, take this readiness quiz.

Simplify: √75.
If you missed this problem, review Example 9.12.

BE PREPARED 10.2

Simplify: √643.
If you missed this problem, review Example 9.67.

BE PREPARED 10.3

Factor: 4x^2−12x+9.
If you missed this problem, review Example 7.43.

Quadratic equations are equations of the form ax^2+bx+c=0 , where a≠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.

We have seen that some quadratic equations can be solved by factoring. In this chapter, we will use three other methods to solve quadratic equations.

Solve Quadratic Equations of the Form ax2 = k Using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x^2=9x^2=9.

We can easily use factoring to find the solutions of similar equations, like x^2=16x^2=16 and x^2=25x^2=25, because 16 and 25 are perfect squares. But what happens when we have an equation like x^2=7x^2=7? Since 7 is not a perfect square, we cannot solve the equation by factoring.

These equations are all of the form x^2=kx^2=k.
We defined the square root of a number in this way:

If n2=m ,then n is a square root ofm

This leads to the Square Root Property.

SQUARE ROOT PROPERTY

Notice that the Square Root Property gives two solutions to an equation of the form x^2=k: the principal square root of kk and its opposite. We could also write the solution as √x=±k.

Now, we will solve the equation x^2=9again, this time using the Square Root Property.

What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x^2=7

EXAMPLE 10.1

Solve: x2=169

How To

Solve a quadratic equation using the Square Root Property.

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

To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.

The Square Root Property started by stating, ‘If x^2=k, and k≥0’. What will happen if k<0? This will be the case in the next example.

Remember, we first isolate the quadratic term and then make the coefficient equal to one.

The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.

Solve Quadratic Equations of the Form a(xh)2 = k Using the Square Root Property

We can use the Square Root Property to solve an equation like (x−3)2=16, too. We will treat the whole binomial, (x−3), as the quadratic term.

Remember, when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.

We will start the solution to the next example by isolating the binomial.

The left sides of the equations in the next two examples do not seem to be of the form a(xh)2. But they are perfect square trinomials, so we will factor to put them in the form we need.

Media

Access these online resources for additional instruction and practice with solving quadratic equations:

  • Solving Quadratic Equations: Solving by Taking Square Roots
  • Using Square Roots to Solve Quadratic Equations
  • Solving Quadratic Equations: The Square Root Method

Section 10.1 Exercises

Practice Makes Perfect

Solve Quadratic Equations of the form ax2=k Using the Square Root Property

In the following exercises, solve the following quadratic equations.

Solve Quadratic Equations of the Form a(x−h)2=k Using the Square Root Property

In the following exercises, solve the following quadratic equations.

Mixed Practice

In the following exercises, solve using the Square Root Property.

Everyday Math


53.Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation 3s2=48 to find s, the length of each garden side.


54.Kathy is drawing up the blueprints for a house she is designing. She wants to have four square windows of equal size in the living room, with a total area of 64 square feet. Solve the equation 4s2=64 to find s, the length of the sides of the windows.

Everyday Math

55.Explain why the equation x2+12=8 has no solution.

56.Explain why the equation y2+8=12 has two solutions.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently: Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help: This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no-I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.