Section 7.1 Extraction of Roots and Properties of Square Roots.
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Transcript of Section 7.1 Extraction of Roots and Properties of Square Roots.
7.1 Lecture Guide: Extraction of Roots and Properties of Square Roots
Objective 1: Solve quadratic equations by extraction of roots.
A quadratic equation is an equation that can be written in the _________________________ form .The quadratic equation can be rewritten as . The two solutions of this equation are the two numbers whose square is 3. We can denote these solutions by One solution is and the other is . The notation “ ” is read plus or minus. This process of solving quadratic equations is called extraction of roots.
2 0ax bx c
3.x 3x 3x
2 3 0x 2 3x
Extraction of Roots Example
If k is a positive real number, then the equation has two real
solutions, :
.
2x kx k
andx k x k
2 3x
3 3x or x
Use extraction of roots to determine the exact solutions of each quadratic equation.
1. 2.2 16x 2 7x
3. 4.22 18x 24 12 0x
Use extraction of roots to determine the exact solutions of each quadratic equation.
Objective 2: Use the product rule for radicals to simplify square roots.
Product Rule for
Square Roots
Algebraic Example
Numerical Example
Ifare both real numbers, then
andx y
.xy x y
12 4 3
4 3
2 3
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
9.2 24x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
10. 2 54x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
11. 22 64x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
12. 25 100 0x
Objective 3: Use the Quotient rule for radicals to simplify square roots.
Quotient Rule for Square Roots
Algebraic Example
Numerical Example
If are both real numbers and , then .
andx y
0y x xy y
25 2536 36
56
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
17. 225 11x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
18. 24 3 0x
The process of rewriting a radical expression so that the _________________ does not have any radicals in it is called rationalizing the denominator. This process uses
the fact that for ,
Objective 4: Simplify expressions of the form by rationalizing the denominator.
a
b
0x 2 .x x x x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
23. 25 9x
Solve each quadratic equation using extraction of roots to obtain exact solutions. Then approximate any irrational solutions to the nearest hundredth.
24. 210 15 0x
In the following box we extend the method of extraction of roots to a quadratic equation that contains the square of a binomial on the left side of the equation.
Extraction of Roots Example
For a positive real number k and nonzero real numbers a
and b: If , then
.
or 2ax b k
ax b k
22 1 3x
2 1 3x 2 1 3x
31. Interest Rate A certificate of deposit earns an annual rate of interest r that is compounded twice a year. An investment of $1,300 grows to $1,412.85 by the end of 1 year. Use the
formula to approximate the annual interest
rate to the nearest tenth of a percent.
2
12r
A P
32.Checking a Solution You can check solutions of any equation by substitution or by graphing.
(a) In problem 18 from earlier in this section you were asked to solve The solutions of this equation are
Check these solutions of this by substitution.
24 3 0.x 3
.2
x