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By the end of this section, you will be able to:
  • Solve quadratic equations of the form a x 2 = k using the Square Root Property
  • Solve quadratic equations of the form a ( x h ) 2 = k using the Square Root Property

Before you get started, take this readiness quiz.

  1. Simplify: 75 .
    If you missed this problem, review [link] .
  2. Simplify: 64 3 .
    If you missed this problem, review [link] .
  3. Factor: 4 x 2 12 x + 9 .
    If you missed this problem, review [link] .

Quadratic equations are equations of the form a x 2 + b x + 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 equation    s 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 ax 2 = 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 = 9 .

x 2 = 9 Put the equation in standard form. x 2 9 = 0 Factor the left side. ( x 3 ) ( x + 3 ) = 0 Use the Zero Product Property. ( x 3 ) = 0 , ( x + 3 ) = 0 Solve each equation. x = 3 , x = −3 Combine the two solutions into ± form. x = ± 3 (The solution is read x is equal to positive or negative 3.’)

We can easily use factoring to find the solutions of similar equations, like x 2 = 16 and x 2 = 25 , because 16 and 25 are perfect squares. But what happens when we have an equation like x 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 = k .
We defined the square root of a number in this way:

If n 2 = m , then n is a square root of m .

This leads to the Square Root Property    .

Square root property

If x 2 = k , and k 0 , then x = k or x = k .

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

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

x 2 = 9 Use the Square Root Property. x = ± 9 Simplify the radical. x = ± 3 Rewrite to show the two solutions. x = 3 , x = −3

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

Use the Square Root Property. x 2 = 7 x = ± 7 Rewrite to show two solutions. x = 7 , x = 7 We cannot simplify 7 , so we leave the answer as a radical.

Solve: x 2 = 169 .

Solution

Use the Square Root Property. Simplify the radical. x 2 = 169 x = ± 169 x = ± 13 Rewrite to show two solutions. x = 13 , x = −13

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Solve: x 2 = 81 .

x = 9 , x = −9

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Solve: y 2 = 121 .

y = 11 , y = −11

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How to solve a quadratic equation of the form a x 2 = k Using the square root property

Solve: x 2 48 = 0 .

Solution

The image shows the given equation, x squared minus 48 equals zero. Step one is to isolate the quadratic term and make its coefficient one so add 48 to both sides of the equation to get x squared by itself. Step two is to use the Square Root Property to get x equals plus or minus the square root of 48. Step three, simplify the square root of 48 by writing 48 as the product of 16 and three. The square root of 16 is four. The simplified solution is x equals plus or minus four square root of three. Step four, check the solutions by substituting each solution into the original equation. When x equals four square root of three, replace x in the original equation with four square root of three to get four square root of three squared minus 48 equals zero. Simplify the left side to get 16 times three minus 48 equals zero which simplifies further to zero equals zero, a true statement. When x equals negative four square root of three, replace x in the original equation with negative four square root of three to get negative four square root of three squared minus 48 equals zero. Simplify the left side to get 16 times three minus 48 equals zero which simplifies further to zero equals zero, also a true statement.
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Solve: x 2 50 = 0 .

x = 5 2 , x = −5 2

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Solve: y 2 27 = 0 .

y = 3 3 , y = −3 3

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Solve a quadratic equation using the square root property.

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

Practice Key Terms 2

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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