MTH55_Lec-48_sec_8-1a_SqRt_Property.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

[email protected] • MTH55_Lec-48_sec_8-1a_SqRt_Property.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§8.1 Complete§8.1 CompleteThe SquareThe Square



Review §Review §

Any QUESTIONS About• §7.7 → Complex Numbers

Any QUESTIONS About HomeWork• §7.7 → HW-30

7.7 MTH 55



The Square Root PropertyThe Square Root Property Let’s consider x2 = 25. We know that the number 25 has

two real-number square roots, 5 and −5, which are the solutions to this equation.

Thus we see that square roots can provide quick solutions for equations of the type x2 = k.



SQUARE ROOT PROPERTYSQUARE ROOT PROPERTY For any nonzero real number d, and

any algebraic expression u, then the Equation u2 = d has exactly two solutions:

dududu orthen If 2

Alternatively in a ShortHand Notation:

dudu then If 2



Example Example Use SqRt Property Use SqRt Property Solve 5x2 = 15. Give exact solutions and

approximations to three decimal places. SOLUTION 25 15x

2 3x

3 or 3.x x

Isolating x2

Using the Property of square roots

The solutions are which round to 1.732 and −1.732.

3Or x ShortHand Notation

3,



Example Example Use SqRt Property Use SqRt Property Solve x2 = 108 SOLN

Check

Use the square root principle.

Simplify by factoring out a perfect square.

2 108x 108x

6 3x

Check

6 3 : 26 3 108

Note: Remember the ± means that the two solutions are and .

6 3 6 3

6 3 : 26 3 108

336x

108336 108336



Example Example Use SqRt Property Use SqRt Property Solve x2 +14 = 32

SOLN 2 14 32x 2 18x

18x

3 2x

29x

Subtract 14 from both sides to isolate x2

Use the square root property

Simplify by factoring out a perfect square



Example Example Use SqRt Property Use SqRt Property Solve (x + 3)2 = 7 SOLN

Using the Property of square roots

The solutions are The check is left for us to do Later

Solving for x

2( 3) 7x

3 7 or 3 7x x

3 7 or 3 7.x x

73Or x ShortHand Notation

3 7.



Example Example Use SqRt Property Use SqRt Property Solve 16x2 + 9 = 0 SOLN

The solutions are The check is left for Later

216 9 0x

2 9/16x

Recall that 1 .i

169or169 xx

ixix43or

43

i43



Solving Quadratic EquationsSolving Quadratic Equations To solve equations in the form ax2 = b, first

isolate x2 by dividing both sides of the equation by a.

Solve an equation in the form ax2 + b = c by using both the addition and multiplication principles of equality to isolate x2 before using the square root principle

In an equation in the form (ax + b)2 = c, notice the expression ax + b is squared. Use the square root principle to eliminate the square.



Example Example Use SqRt Property Use SqRt Property Solve (5x − 3)2 = 4 SOLN

Add 3 to both sides and divide each side by 5, to isolate x.

Use the square root property

25 3 4x

5 3 4x

or

5 3 2x 2 35

x

2 35

x

2 35

x

1x 15

x or



Completing the SquareCompleting the Square Not all quadratic equations can be

solved as in the previous examples. By using a method called

completing the square, we can use the principle of square roots to solve any quadratic equation

To Complete-the-Sq we Add ZERO to an expression or equation



Example Example Complete the Sq Complete the Sq Solve x2 + 10x + 4 = 0 SOLN:

2( 5) 21x

2 10 4 0x x 2 10 4x x

x2 + 10x + 25 = –4 + 25

5 21 or 5 21x x

5 21.

Using the property of square roots

Factoring

Adding 25 to both sides.

The solutions are The check is left for Later



Solving Quadratic Equations Solving Quadratic Equations by Completing the Squareby Completing the Square Write the equation in the form 11·x2 + bx = c. Complete the square by adding (b/2)2 to both

sides.• (b/2)2 is called the “Quadratic Supplement”

Write the completed square in factored form. Use the square root property to eliminate the

square. Isolate the variable. Simplify as needed.



Example Example Complete the Sq Complete the Sq Solve by Completing the Square:

2x2 − 10x = 9 SOLN:

Divide both sides by 2.

2 952

x x Simplify.

Add to both sides to complete the square.

22 10 9x x 22 10 9

2 2x x

2 25 9 2554 2 4

x x 254



Example Example Complete the Sq Complete the Sq Solve 2x2 − 10x = 9 SOLN:

Combine the fractions.

Factor.

Add to both sides and simplify the square root.

52

Use the square root principle.

443

425

418

25 2

x

443

25

x

243

25

x

2435

x



Example Example Complete the Sq Complete the Sq Solve by Completing the Square:

3x2 + 7x +1 = 0 SOLUTION: The coefficient of the x2

term must be 11. When it is not, multiply or divide on both sides to find an equivalent eqn with an x2 coefficient of 1.

23 7 1 0x x

1 13 3

23 7 1 0 x x Divide Eqn by 3



Example Example Complete the Sq Complete the Sq Solve: 3x2 + 7x +1 = 0 SOLN: 2 7 1 0

3 3x x

2 7 1 3 3

x x

2 49 4936

7 13 3 36

x x

27 12 496 36 36

x



Example Example Complete the Sq Complete the Sq Solve: 3x2 + 7x +1 = 0 SOLN:

7 37 7 37 or 6 6 6 6

x x

27 376 36

x

7 37 7 37 or 6 6 6 6

x x

27 376 36

x

Square Root Property

Isolatex

Taking the Square Root of Both Sides



Example Example Taipei 101 Tower Taipei 101 Tower The Taipei 101 tower in Taiwan is 1670

feet tall. How long would it take an object to fall to the ground from the top?

Familiarize: A formula for Gravity-Driven FreeFall with negligible air-drag is s = 16t2 • where

– s is the FreeFall Distance in feet – t is the FreeFall Time in seconds



Example Example Taipei 101 Tower Taipei 101 Tower Translate: We know the distance is

1670 feet and that we need to solve for tSub 1670 for s → 1670 = 16t2

CarryOut: 1670 = 16t2

2167016

t

1670 1670 or 16 16

t t

10.2 or 10.2 t t



Example Example Taipei 101 Tower Taipei 101 Tower Check: The number −10.2 cannot be a

solution because time cannot be negative.• Check t = 10.2 in formula:

s = 16(10.2)2 = 16(104.04) = 1664.64– This result is very close to the 1670 value.

State. It takes about 10.2 seconds for an object to fall to the ground from the top of the Taipei 101 tower.



Compound InterestCompound Interest After one year, an amount of money P,

invested at 4% per year, is worth 104% of P, or P(1.04). If that amount continues to earn 4% interest per year, after the second year the investment will be worth 104% of P(1.04), or P(1.04)2. This is called compounding interest since after the first period, interest is earned on both the initial investment and the interest from the first period. Generalizing, we have the following.



Compound Interest FormulaCompound Interest Formula If an amount of money P is

invested at interest rate r, compounded annually, then in t years, it will grow to the amount A as given by the Formula

• Note that r is expressed as a DECIMAL



Example Example Compound Interest Compound Interest Tariq invested $5800 at an interest rate

of r, compounded annually. In two years, it grew to $6765.

What was the interest rate? Familiarize: This is a compound

interest calculation and we are already familiar with the compound-interest formula.



Example Example Compound Interest Compound Interest Translate The translation consists of

substituting into the Interest formula

(1 )tA P r 6765 = 5800(1 + r)2

CarryOut: Solve for r6765/5800 = (1 + r)2

6765/5800 1 r

1 6765/5800 r

.08 or 2.08r r



Example Example Compound Interest Compound Interest Check: Since the interest rate can NOT

negative, we need only to check 0.08 or 8%.

If $5800 were invested at 8% compounded annually, then in 2 yrs it would grow to 5800·(1.08)2, or $6765. • The number 8% checks.

State: Tariq’s interest rate was 8%.



Solving FormulasSolving Formulas Recall that to solve a formula for a

certain letter-variable, we use the principles for solving equations to isolate that letter-variable alone on one side of the Equals-Sign• The Bernoulli

Equation for an InCompressibleFluid:

CgPz

gV

2

2



Example Example Solve Solve 21 5 for .2

D n n n

SOLN 21 52

D n n

22 5D n n Multiplying both sides by 2

Complete the Square

Express LHS as Perfect Square

222

252

255

Dnn

4258

4252

25 2

DDn

Solve Using Square Root Principle2

2585

Dn



Solve a Formula for a Letter – Say, Solve a Formula for a Letter – Say, bb1. Clear fractions and use the principle of powers,

as needed. Perform these steps until radicals containing b are gone and b is not in any denominator.

2. Combine all like terms.3. If the only power of b is b1, the equation can be

solved without using exponent rules.4. If b2 appears but b does not, solve for b2 and

use the principle of square roots to solve for b.5. If there are terms containing both b and b2,

put the equation in standard form and Complete the Square.



WhiteBoard WorkWhiteBoard Work

Problems From §8.1 Exercise Set• 22, 44, 56, 78, 88

Solve ax2 + bx + c = 0 by completing the square:



All Done for TodayAll Done for Today

Taipei 101Tower

Taipei, R.o.C.



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



Graph Graph yy = | = |xx|| Make T-table

x y = |x |-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



-3

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls -5

-4

-3

-2

-1

0

1

2

3

4

5

-10 -8 -6 -4 -2 0 2 4 6 8 10

M55_§JBerland_Graphs_0806.xls

x

y

MTH55_Lec-48_sec_8-1a_SqRt_Property.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical.

Documents