3.6 solving quadratic equations by finding the square root
TRANSCRIPT
3.6 Solving Quadratic Equations
By Finding the Square Root
Simplifying Expressions with
Radicals Perfect Square can’t be in radical Fraction can’t be in radical Radical can’t be in the denominator
Rationalize the denominator To add/subtract, you must have like
radicands
EXAMPLE 1 Use properties of square roots
Simplify the expression.
5= 4a.
80 516=
= 3 14b.
6 21 126= 9 14=
c.
481 =
4
81=
2 9
d.
716 =
7
16=
47
GUIDED PRACTICEGUIDED PRACTICE
271.
3 3
982.
2 7
ANSWER
ANSWER
3.
6 5
10 15
4. 8 28
14 4
ANSWER
ANSWER
5.
3 8
964
6. 154
215ANSWER
ANSWER
7. 1125
511 8. 36
49 76
EXAMPLE 2 Rationalize denominators of fractions.
Simplify (a) 52
and 3
7 + 2
SOLUTION
(a) 52
210=
=5
2
=5
2
2
2
(b)
(b) 3
7 + 2=
3
7 + 2 7 – 2
7 – 2
= 21 – 3 2
49 – 7 + 7 – 2 2 2
= 21 – 3 247
Key to solving using square roots
Remember—You have to include the positive (principle) root as well as the negative root.
x2 = 121 x = ± 11
EXAMPLE 3 Solve a quadratic equation
Solve 3x2 + 5 = 41.
3x2 + 5 = 41 Write original equation.
3x2 = 36 Subtract 5 from each side.
x2 = 12 Divide each side by 3.
x =
+ 12 Take square roots of each side.
x =
+ 4 3
x =
+ 2 3
Product property
Simplify.
EXAMPLE 3 Solve a quadratic equation
ANSWER
The solutions are and 2 3 2 3–
Check the solutions by substituting them into the original equation.
3x2 + 5 = 413( )2 + 5 = 412 3
?
41 = 413(12) + 5 = 41
?
3x2 + 5 = 413( )2 + 5 = 41 – 2 3
?
41 = 413(12) + 5 = 41
?
EXAMPLE 4 Standardized Test Practice
SOLUTION
15
(z + 3)2 = 7 Write original equation.
(z + 3)2 = 35 Multiply each side by 5.
z + 3 = + 35 Take square roots of each side.
z = –3 + 35 Subtract 3 from each side.
The solutions are –3 + and –3 – 35 35
GUIDED PRACTICEGUIDED PRACTICE
Simplify the expression.
9.
530
65
10. 98
24
3
11. 1712
516
ANSWER
ANSWER
ANSWER
12.
39921
1921
– 21 – 3 522
13. – 6
7 – 5
8 – 2 115
14. 2
4 + 11
ANSWER
ANSWER
ANSWER
– 9 + 774
GUIDED PRACTICE
15. – 1
9 + 7
32 + 4 361
16. 4
8 – 3
ANSWER
ANSWER
GUIDED PRACTICE
17.
Solve the equation.
5x2 = 80
+ 4
18. z2 – 7 = 29
+ 6
19. 3(x – 2)2 = 40
120
32 +
ANSWER
ANSWER
ANSWER