10.1: Solve Quadratic Equations Using the Square Root Property
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- May 3, 2019
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Learning Objectives
By the end of this section, you will be able to:
- Solve quadratic equations of the form ax2=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.
- Simplify: √75.
- Simplify: √643
- Factor: 4x2−12x+9.
Quadratic equations are equations of the form ax2+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 x2=9.
x2=9Put the equation in standard form.x2−9=0Factor the left side.(x−3)(x+3)=0Use the Zero Product Property.(x−3)=0,(x+3)=0Solve each equation.x=3,x=−3Combine the two solutions into±formx=±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 x2=16 and x2=25, because 16 and 25 are perfect squares. But what happens when we have an equation like x2=7? Since 7 is not a perfect square, we cannot solve the equation by factoring.
These equations are all of the form x2=k.
We defined the square root of a number in this way:
If n2=m, then n is a square root of m.
This leads to the Square Root Property.
Definition: SQUARE ROOT PROPERTY
If x2=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 x2=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 x2=9 again, this time using the Square Root Property.
x2=9Use the Square Root Property.x=±√9Simplify the radical.x=±3Rewrite 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 x2=7.
Use the Square Root Property. x=±√7Rewrite to show two solutions.x=√7,x=−√7We cannot simplify√7 so we leave the answer as a radical.
Example 10.1.1
Solve: x2=169
- Answer
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x2=169Use the Square Root Property.x=±√169Simplify the radical.x=±13Rewrite to show two solutions.x=13,x=−13
Example 10.1.2
Solve: x2=81
- Answer
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x=9, x=−9
Example 10.1.3
Solve: y2=121
- Answer
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y = 11, y = −11
How to Solve a Quadratic Equation of the Form ax2=k Using the Square Root Property
Example 10.1.4
Solve: x2−48=0
- Answer
Example 10.1.5
Solve: x2−50=0
- Answer
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x=5√2,x=−5√2
Example 10.1.6
Solve: y2−27=0
- Answer
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y=3√3,x=−3√3
Definition:SOLVE A QUADRATIC EQUATION USING THE SQUARE ROOT PROPERTY.
- Isolate the quadratic term and make its coefficient one.
- Use Square Root Property.
- Simplify the radical.
- 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.
Example 10.1.7
Solve: 5m2=80
- Answer
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The quadratic term is isolated. 5m2=80 Divide by 5 to make its cofficient 1. 5m25=805 Simplify. m2=16 Use the Square Root Property. m=±√16 Simplify the radical. m=±4 Rewrite to show two solutions. m=4,m=−4 Check the solutions.
Example 10.1.8
Solve: 2x2=98.
- Answer
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x=7, x=−7
Example 10.1.9
Solve: 3z2=108.
- Answer
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z=6, z=−6
The Square Root Property started by stating, ‘If x2=k, and k≥0’. What will happen if k<0? This will be the case in the next example.
Example 10.1.10
Solve: q2+24=0.
- Answer
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q2=24Isolate the quadratic term.q2=−24Use the Square Root Property.q=±√−24The√−24is not a real numberThere is no real solution
Example 10.1.11
Solve: c2+12=0.
- Answer
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no real solution
Example 10.1.12
Solve: d2+81=0.
- Answer
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no real solution
Example 10.1.13
Solve: 23u2+5=17.
- Answer
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23u2+5=17 Isolate the quadratic term. 23u2=12
Multiply by 32 to make the coefficient 1. 32·23u2=32·12 Simplify. u2=18 Use the Square Root Property. u=±√18 Simplify the radical. u=±√9√2 Simplify. u=±3√2 Rewrite to show two solutions. u=3√2, u=−3√2 Check.
Example 10.1.14
Solve: 12x2+4=24
- Answer
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x=2√10, x=−2√10
Example 10.1.15
Solve: 34y2−3=18.
- Answer
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y=2√7, y=−2√7
The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.
Example 10.1.16
Solve: 2c2−4=45.
- Answer
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2c2−4=45 Isolate the quadratic term. 2c2=49 Divide by 2 to make the coefficient 1. 2c22=492 Simplify. c2=492 Use the Square Root Property. c=±√49√2 Simplify the radical. c=±√49√2 Rationalize the denominator. c=±√49√2√2√2 Simplify. c=±7√22 Rewrite to show two solutions. c=7√22, c=−7√22 Check. We leave the check for you.
Example 10.1.17
Solve: 5r2−2=34.
- Answer
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r=6√55, r=−6√55
Example 10.1.18
Solve: 3t2+6=70.
- Answer
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t=8√33,t=−8√33
Solve Quadratic Equations of the Form a(x−h)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.
Example 10.1.19
Solve: (x−3)2=16.
- Answer
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(x−3)2=16 Use the Square Root Property. x−3=±√16 Simplify. x−3=±4 Write as two equations. x−3=4, x−3=−4 Solve. x=7,x=−1 Check.
Example 10.1.20
Solve: (q+5)2=1.
- Answer
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q=−6, q=−4
Example 10.1.21
Solve: (r−3)2=25.
- Answer
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r=8, r=−2
Example 10.1.22
Solve: (y−7)2=12.
- Answer
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(y−7)2=12. Use the Square Root Property. y−7=±√12 Simplify the radical. y−7=±2√3 Solve for y. y=7±2√3 Rewrite to show two solutions. y=7+2√3,y=7−2√3 Check.
Example 10.1.23
Solve: (a−3)2=18.
- Answer
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a=3+3√2, a=3−3√2
Example 10.1.24
Solve: (b+2)2=40.
- Answer
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b=−2+2√10, b=−2−2√10
Example 10.1.25
Solve: (x−12)2=54.
- Answer
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(x−12)2=54 Use the Square Root Property. (x−12)=±√54 Rewrite the radical as a fraction of square roots. (x−12)=±√5√4 Simplify the radical. (x−12)=±√52 Solve for x. x=12+±√52 Rewrite to show two solutions. x=12+√52, x=12−√52 Check. We leave the check for you
Example 10.1.26
Solve: (x−13)2=59.
- Answer
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x=13+√53, x=13−√53
Example 10.1.27
Solve: (y−34)2=716.
- Answer
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y=34+√74, y=34−√74,
We will start the solution to the next example by isolating the binomial.
Example 10.1.28
Solve: (x−2)2+3=30.
- Answer
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(x−2)2+3=30 Isolate the binomial term. (x−2)2=27 Use the Square Root Property. x−2=±√27 Simplify the radical. x−2=±3√3 Solve for x. x=2+±3√3 x−2=±3√3 x=2+3√3, x=2−3√3 Check. We leave the check for you
Example 10.1.29
Solve: (a−5)2+4=24.
- Answer
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a=5+2√5, a=5−2√5
Example 10.1.30
Solve: (b−3)2−8=24.
- Answer
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b=3+4√2, b=3−4√2
Example 10.1.31
Solve:(3v−7)2=−12.
- Answer
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(3v−7)2=−12 Use the Square Root Property. 3v−7=±√−12 The √−12 is not a real number. There is no real solution.
Example 10.1.32
Solve: (3r+4)2=−8.
- Answer
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no real solution
The left sides of the equations in the next two examples do not seem to be of the form a(x−h)2. But they are perfect square trinomials, so we will factor to put them in the form we need.
Example 10.1.33
Solve: p2−10p+25=18.
- Answer
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The left side of the equation is a perfect square trinomial. We will factor it first.
p2−10p+25=18 Factor the perfect square trinomial. (p−5)2=18 Use the Square Root Property. p−5=±√18 Simplify the radical. p−5=±3√2 Solve for p. p=5±3√2 Rewrite to show two solutions. p=5+3√2, p=5−3√2 Check. We leave the check for you.
Example 10.1.34
Solve: x2−6x+9=12.
- Answer
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x=3+2√3, x=3−2√3
Example 10.1.35
Solve: y2+12y+36=32.
- Answer
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y=−6+4√2, y=−6−4√2
Example 10.1.36
Solve: 4n2+4n+1=16.
- Answer
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Again, we notice the left side of the equation is a perfect square trinomial. We will factor it first.
4n2+4n+1=16 Factor the perfect square trinomial. (2n+1)2=16 Use the Square Root Property. (2n+1)=±√16 Simplify the radical. (2n+1)=±4 Solve for n. 2n=−1±4 Divide each side by 2. 2n2=−1±42
n=−1±42
Rewrite to show two solutions. n=−1+42, n=−1−42 Simplify each equation. n=32, n=−52 Check.
Example 10.1.37
Solve: 9m2−12m+4=25.
- Answer
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m=73, m=−1
Example 10.1.38
Solve: 16n2+40n+25=4.
- Answer
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n=−34, n=−74
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
Key Concepts
- Square Root Property
If x2=k, and k≥0, then x=√k or x=−√k.
Glossary
- quadratic equation
- A quadratic equation is an equation of the form ax2+bx+c=0 where a≠0.
- Square Root Property
- The Square Root Property states that, if x2=k, and k≥0, then x=√k or x=−√k.