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10.5
Transcript of 10.5
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.
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