6.6 Quadratic Equations

Multiplying Binomials

• A binomial has 2 terms

• Examples: x + 3, 3x – 5, x2 + 2y2, a – 10b

• To multiply binomials use the FOIL method

(x + 3)(x + 4) = First

OuterInner

Last

24 3 3 4 7 12x x x x x x First

LastOuter Inner

Examples:Multiply

• (x + 5)(x + 6)

• (3x – 4)( 5x + 3)

Common factors

• When factoring polynomials, first look for a common factor in each term

• Example: The binomial below has the factor 3 in each term

3x + 6y = (3)x + (3)2y = 3(x + 2y)

To factor the above polynomial we used the distributive property.

ac + bc = c(a + b)

Examples: Factoring by distributive property

1. 16n2 + 12n

2. 4x2 +20x -12

Examples: Difference of two squares a2 –b2 = (a + b)(a – b)

1. x2 – 16

2. 9y2 - 25

3. 49x2 – 36 z2

4. x4 – 81

5. 4x2y2 – b4

6. 3x3 – 12x

Factoring Trinomials

• A trinomial has three terms.

• Example: x2 + 5x + 6

• If a trinomial factors, it factors into two binomials.

Factoring trinomials

• First, look for a common factor.

• Then look for perfect square trinomials:

a2 + 2ab + b2 = (a + b)2

a2 – 2ab + b2 = (a – b)2

• If it is not a perfect square trinomial then factor into two binomials.

Examples: Perfect square trinomialsa2 + 2ab + b2 = (a + b)2

a2 – 2ab + b2 = (a – b)2

1. x2 + 10x + 25

2. x2 – 8y + 16

3. 9x2 – 24x + 16

4. 25x2 + 80xy + 64y2

5. 8x2y – 24xy + 18y

Factoring trinomials of the form x2 + bx + c

• x2 + bx + c = (x + ___)(x + ____)

• x2 - bx + c = (x - ___)(x - ____)

• To fill in the blanks look for factors of c that add up to equal b

Examples: Factor. Check answers by FOIL

• x2-5x+6

• x2+6x+8

• x2-7x+10

• x2+7x+12

Factoring trinomials of the form x2 + bx - c

• x2 + bx - c = (x + ___)(x - ____)

• To fill in the blanks look for factors of c that subtract to equal b

• If the 1st sign is negative place the larger factor with the negative sign

• If the 1st sign is positive place the larger factor with the positive sign

Signs will be different

Examples: Factor

• x2+2x-35

• x2-4x-12

• x2-2x+15

• x2+5x-36

Factoring trinomials of the form ax2 + bx + c

• One method is trial and error

• Try factors of a and c then FOIL to see if it works

• Examples:

• 2x2 + 15x + 28 = ( + )( + )

• 3x2 +7x – 20= ( + )( - )

Alternative method: Factor by grouping

• Factor by grouping is used to factor polynomials with 4 terms

• Example: Factor 10x2 – 15x + 4x – 6

(10x2 – 15x) + (4x – 6)

5x(2x – 3) + 2(2x – 3)

(2x – 3)(5x + 2)

Group together 1st 2 terms and last 2 terms

Factor out any common factors in each group

Factor out (2x – 3) from each term

Examples: Factor by grouping

Factoring trinomials using factor by grouping.

• Since factor by grouping involves 4 terms we want to rewrite the trinomial as a polynomial with 4 terms

General trinomials: ax2 + bx + c

• Example: 2x2 + 5x – 3• Multiply ac = 2(-3) = -6

• Select 6 and -1

Factors of –6 Sum of factors

1, -6 -5

2, -3 -1

3, - 2 1

6, -1 5*

Example cont’d

2x2 + (___ + ___) – 3

2x2 + (6x + -1x) – 3

(2x2 + 6x) + (-x – 3)

2x(x + 3) + -1(x + 3)

(x + 3)(2x – 1)

More examples

• 3x2 – 14x – 5

• 15m2 + 14m – 8

• 12x2 + 23p + 5

• 12 – 20x – 13x2

Zero Product Property

• To solve a quadratic equation by factoring we will use the zero product property:

If ab = 0,

then a = 0 or b = 0

where a and b are any real numbers.

Examples: Solve by factoring

1. x2 – 3x – 28 = 0

2. x2 + 4x = 12

More examples

3. 16x2 = 49

4. 4x2 – 35x – 5 = 4

More examples

5. x2 + 6x = 13 = 4

6. 4x2 – 12x = 0

Examples: Solve by factoring

1. x2 – 3x – 28 = 0

2. x2 + 4x = 12

3. 4x2 – 35x – 5 = 4

4. 4x2 – 12x = 0

Examples: Solve by finding square roots

1. 16x2 = 49

2. 5x2 – 180 = 0

3. 3x2 = 24

4. x2 – ¼ = 0

Quadratic formula

• To use the quadratic formula, the equation must be in the form ax2 + bx + c = 0

2 4

2

b b acx

a

Examples: solve using the quadratic formula

1. 2y2 + 4y = 30

2. x2 – 7x + 1 = 0

3. 5m2 + 7m = -3

4. x2 + 16 = 8x

Discriminant

• The discriminant is b2 – 4ac.

• The following chart describes the roots based on the value of the discriminant.

b2 – 4ac Roots Graph

> 0 2 real Intersects x-axis twice

< 0 2 imaginary Does not intersect x-axis

= 0 1 real Intersects x-axis once

Examine the determinants in the previous examples to verify the

chart