Group6Chapter6Quadratic Equations

7/31/2019 Group6Chapter6Quadratic Equations

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QUADRATIC EQUATIONS


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OBJECTIVES:

Find the sum, difference, productand quotient of two complexnumbers.

Solve quadratic equations. Determine and discuss the nature

of the roots of quadratic equations

Find the sum and the product of theroots of quadratic

Solve equations in quadratic forms


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Solve radical equations

Discuss the graphs of quadratic forms

Solve systems involving quadraticequations by the graphical and algebraic

methods

Apply the concepts involved in quadratic

equations to actual life situations


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WHATARE QUADRATIC EQUATIONS?

In mathematics, a quadratic equation is a polynomial equationof the second degree. The general form is:

wherexrepresents a variable or an unknown, and a, b,and c are constants with a 0.

General Form


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COMPLEXNUMBERS

A complex number is a number of the form a+ bi, where a and b are real numbers and iisthe imaginary unit

Quadratic Equations

Complex NumbersSolving QuadraticEquations by FactoringSolving QuadraticEquations byCompleting the SquareSolving QuadraticEquations by theQuadratic Formula

Nature of the roots ofQuadratic EquationsSum and Product ofthe Roots Equations inQuadratic Form Radical Equations Fractional Equations Quadratic Curves Systems involvingQuadratic Equations ApplicationsQuadratic Equations

a+bi
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You can't take the square root of a negative number. That's because

you had no numbers which were negative after you'd squared them.

Every number was positive after you squared it. So you couldn't

very well square-root a negative and expect to come up with

anything sensible.

You can take the square root of a negative number, but it involves

using a new number to do it. This new number was invented around

the time of the Reformation. At that time, nobody believed that any

"real world" use would be found for this new number, other thaneasing the computations involved in solving certain equations, so

the new number was viewed as being a pretend number invented

for convenience sake.

Complex NumbersQuadratic Equations




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This new number was called "i", standing for"imaginary", because "everybody knew" that iwasn't"real". (That's why you couldn't take the square root of anegative number before: you only had "real" numbers;that is, numbers without the "i" in them.) The imaginaryis defined to be:

Quadratic Equations




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NOW YOU MAY THINK:

This is Wrong

Quadratic Equations




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This pattern of powers, signs, 1's, and i's is a cycle:

In other words, to calculate any high power ofi, you canconvert it to a lower power by taking the closestmultiple of 4 that's no bigger than the exponent andsubtracting this multiple from the exponent.

Quadratic Equations




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EXAMPLES :

1. Simplify sqrt(9).



4. Simplify

sqrt(

6).

5. Simplify 2i+ 3i.

6. Simplify 16i 5i.

7. Multiply and simplify (3i)(4i). 8. Multiply and simplify (i)(2i)(3i).

9. Simplify i17.

10. Simplify i120.

Quadratic Equations




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SUM, D IFFERENCEAND PRODUCTOF

COMPLEX NUMBERS

1. To add/subtract two complex numbers, take the sum/difference ofthe real parts and the sum/difference of the imaginary parts.

(a) (3+4i) + (7+5i)= (3+7) + (4+5)i= 10+9i

(b) (3+-25) + (2+-16)= (3+5i) + (2+4I)= (3+2) + (5+4)I =5+9i

(c) (10+5i) (-4-3i)= [10-(-4)] = ;5-(-3)i= (10+4) + (5+3)i= 14+8i

2. To find the product of two complex numbers, expand theexpression as the product of two binomials , replace with -1, andsimplify.

(7-4i)(7=4i)= (7)(7) + (7)(4i) + (-4i)(7) = (-4i)(7) + (-4i)(4i)

=49 + 28i-28i-16i2

=49 + 16

=65

Quadratic Equations




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QUOTIENTOF COMPLEX NUMBERS

To find the quotient of two complex numbers, multiply bothnumerator and denominator by the conjugate of thedenominator and then simplify.

Answer

Quadratic Equations




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SOLUTIONS:

2I+ 3I = (2 + 3)I = 5I

1.

2.

3.

4.

5.

Quadratic Equations




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6.

7.

8.

9.

10.

16I 5I = (16 5)I = 11I

(3I)(4I) = (3 4)(II) = (12)(I2)= (12)(1) = 12

(I

)(2I

)(

3I

) = (2

3)(I

I

I

)=(6)(I2 I) =(6)(1 I)

= (6)(I) = 6I

I

17

= I16 + 1

=I4 4 + 1

= I1

=I

I120 = I4 30 = I4 30 + 0 = I0 = 1

Quadratic Equations




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Solving Quadratic Equations By Factoring

If the left side of a complete quadratic equation instandard form can be factored into two linear factors,the roots or solutions of the quadratic equation can beobtained quickly and easily. The key principle involved

in this method rests on the following property of realnumbers.

The Zero Property Of Equality

The product AB = 0, if and only A = 0or B = 0

Quadratic Equations




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Solvex2 + 5x+ 6 = 0.

x2 + 5x+ 6 = (x+ 2)(x+ 3) Factorx2 + 5x+ 6

(x+ 2)(x+ 3) = 0 Set it equal to zero

x+ 2 = 0 or x+ 3 = 0 Solve each factorx= 2 or x= 3

Checking:

x2 + 5x+ 6 = 0:

[3]2 + 5[3] + 6 ?=? 09 15 + 6 ?=? 09 + 6 15 ?=? 0

15 15 ?=? 00 = 0

[2]2 + 5[2] + 6 ?=? 0

4 10 + 6 ?=? 04 + 6 10 ?=? 010 10 ?=? 0

0 = 0

Quadratic Equations




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Solve x 2 = x - 6=6

x 2 + x - 6 = 6

x 2 + x - 12 = 0

(x + 4)(x - 3)=0

x + 4= 0 or x - 3=0

x= -4 or x= 3

Solve 3 x 2 = 2x

3 x 2 =2x

3 x 2 - 2x= 0

x(3x 2)= 0

x=0 or 3x 2= 0

x= 0 or x=

Quadratic Equations




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Solve x2 + 3x= 0

x(x+ 3)=0 Factor the common monomial.

x= 0 or x+ 3 = 0 Set each factor equal to 0 andsolve forx.

x= -3

Solve y2 = 16

y2 - 16 = 0 Get all terms on the same side.

(y+ 4)(y- 4) =0 Factor the difference ofsquares.

y+ 4 = 0 OR y- 4 = 0 Set each factor equal to 0 andsolve fory.y= -4 OR y= 4

Quadratic Equations




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Solve c 2 - 12 = c

c 2 - 12 - c =0 c 2 - c - 12 = 0

(c + 3)(c - 4) = 0c + 3 = 0 OR c - 4 = 0c = -3 OR c = 4

Solve (x 3)(x 4) = 0.x 3 = 0 or x 4 = 0x = 3 or x = 4

Checking x = 3 in (x 3)(x 4) = 0:

([3] 3)([3] 4) ?=? 0(3 3)(3 4) ?=? 0

(0)(1) ?=? 00 = 0

Checking x = 4 in (x 3)(x 4) = 0:

([4] 3)([4] 4) ?=? 0(4 3)(4 4) ?=? 0

(1)(0) ?=? 00 = 0

Quadratic Equations




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Solve x 2 - 3x = 0

x 2 - 3x = 0

x (x - 3) = 0

x = 0 or x - 3 = 0

x = 0

or

x = 3

Solve 2 x 2 + x - 21 = 0

2 x 2 + x - 21 = (2x + a)(x + b)

2 x 2 + x - 21 = (2x + 7)(x - 3)

(2x + 7)(x - 3) = 0

2x + 7 = 0

x - 3 = 0

x = - 7 / 2

or

x = 3

Quadratic Equations




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Solve (x - 1)(x + 1 / 2) = - x + 1

(x - 1)(x + 1 / 2) = - (x - 1)

(x - 1)(x + 1 / 2) + (x - 1) = 0

(x - 1)(x + 1 / 2 + 1) = 0

x - 1 = 0

x + 3 / 2 = 0

x = 1

or

x = - 3 / 2

Quadratic Equations




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Solving Quadratic Equations by Completing the Square

28 20 0x x

2

8 20x x

Solve the following equation by completing the square:

Step 1: Move quadratic term, and linear term to left sideof the equation

Quadratic Equations




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Step 2: Find the term that completes the square on theleft side of the equation. Add that term to both sides

2 8 =20 +x x

21 ( ) 4 then square it, 4 162

8

2

8 2016 16x x

Quadratic Equations




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2 8 2016 16x x

2( 4)( 4) 36

( 4) 36

x x

x

Step 3: Factor the perfect square trinomial on the left sideof the equation. Simplify the right side of theequation.

Quadratic Equations

Complex NumbersSolving Quadratic

Equations by FactoringSolving QuadraticEquations byCompleting the SquareSolving QuadraticEquations by theQuadratic Formula

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Group6Chapter6Quadratic Equations