10.5 Use Square roots to Solve Quadratic Equations

Solving To use square roots to solve a quadratic

equation of the form , first isolate on one side of the equation to obtain . Then use the following:

If , then has two solutions:

If , then has one solution:

If , then has no solution.

Example 1 Solve quadratic equations

Solve the equation.

8=2x2 18=18–m2 – 5=12b2 +a. b. c.

4=x2 Divide each side by 2.

a. Write original equation.8=2x2

=x 4+– Take square roots of each side.

=x 2+– Simplify.

ANSWER The solutions are 2 and 2.–

Example 1 Solve quadratic equations

Add 18 to each side.0=m2

0=m The square root of 0 is 0.

ANSWER The solution is 0.

b. 18–m2 Write original equation.18= –

ANSWER

Negative real numbers do not have real square roots. So, there is no solution.

12b2 +c. 5= Write original equation.

7=b2 – Subtract 12 from each side.

Example 2 Take square roots of a fraction

Solve 9.=4z2

Divide each side by 4.=z2

4

9

Simplify.=z2

3+–

SOLUTION

9=4z2Write original equation.

=z +– 4

9Take square roots of each side.

Example 2 Take square roots of a fraction

ANSWER The solutions are and2

3–

2

3.

Example 3 Solve a quadratic equation

24=3x2 Add 11 to each side.

=x2 8 Divide each side by 3.

Write original equation.13=

SOLUTION

– 113x2

Simplify.=x +– 22

Solve – 13.=113x2

=x +– 8 Take square roots of each side.

ANSWER The solutions are – 22 2.2and

Example 4 Solve a quadratic equation

Solve ( )24x6 – 42.=

ANSWER The solutions are and 4 7.–4 7+

( )24x6 – 42= Write original equation.

( )24x – 7= Divide each side by 6.

=4–x 7–+ Take square roots of each side.

= 4x 7–+ Add 4 to each side.

Example 5

During an ice hockey game, a remote-controlled blimp flies above the crowd and drops a numbered table-tennis ball. The number on the ball corresponds

to a prize. Use the information in the diagram to find the amount of time that the ball is in the air.

SPORTS EVENT

Solve a multi-step problem

Solve a multi-step problemExample 5

SOLUTION

STEP 1Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of the time t (in seconds) after it is dropped.

Find the amount of time the ball is in the air by substituting 17 for h and solving for t.

STEP 2

Vertical motion model16t2h = svt+ +–

Substitute for v and s.16t2h = 450t+ +–

Example 5 Solve a multi-step problem

ANSWER The ball is in the air for about 1.32 seconds.

Substitute 17 for h in model.16t217 = 45+–

Subtract 45 from each side.= 16t2–28–

Divide each side by –16.t2=16

28

t=16

28=

2

7 Take positive square root.

≈ t1.32 Use a calculator.

10.5 Warm-UpSolve the equation.1.

2.

3.

4.

5.