§ 8.2 the quadratic formula. blitzer, intermediate algebra, 5e – slide #2 section 8.2 the...

§ 8.2

The Quadratic Formula

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 8.2


The Quadratic FormulaThe solutions of a quadratic equation in standard form

with , are given by the quadratic formula

02 cbxax 0a

.2

42

a

acbbx

See page 576 of your textbook to see how the quadratic formula is derived using ‘completing the square’.



EXAMPLEEXAMPLE

Solve using the quadratic formula:

SOLUTIONSOLUTION

The given equation is in standard form. Begin by identifying the values for a, b, and c.

.01582 xx

01582 xx

a = 1 b = 8 c = 15

Substituting these values into the quadratic formula and simplifying gives the equation’s solutions.



CONTINUECONTINUEDD

a

acbbx

2

42 Use the quadratic formula.

12

151488 2

xSubstitute the values for a, b, and c: a = 1, b = 8, c = 15.

2

60648 Simplify.

2

48 Subtract.

2

28 The square root of 4 is 2.



CONTINUECONTINUEDD

2

28x

The solutions are -3 and -5. The solution set is {-3,-5}.

Now we will evaluate this expression in two different ways to obtain the two solutions. On the left, we will add 2 to -8. On the right, we will subtract 2 from -8.

2

28 x

32

6

x 5

2

10

x



EXAMPLEEXAMPLE

Solve using the quadratic formula:

SOLUTIONSOLUTION

The quadratic equation must be in standard form to identify the values for a, b, and c. To move all terms to one side and obtain zero on the right, we subtract -4x + 5 from both sides. Then we can identify the values for a, b, and c.

.542 2 xx

a = 2 b = 4 c = -5

542 2 xx This is the given equation.0542 2 xx Subtract -4x + 5 from

both sides.



CONTINUECONTINUEDD

a

acbbx

2

42 Use the quadratic formula.

22

52444 2

xSubstitute the values for a, b, and c: a = 2, b = 4, c = -5.

4

40164 Simplify.

4

564 Add.

Substituting these values into the quadratic formula and simplifying gives the equation’s solutions.



CONTINUECONTINUEDD

4

1424 14214414456

4

1422 Factor out 2 from the numerator.

2

142

Divide the numerator and denominator by 2.

The solutions are , and the solution set is2

142 .2

142

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

• The relationships among the various sets of numbers.


Quadratic Formula – The Discriminant

The Discriminant and

the Kinds of Solutions to TDiscriminant Kinds of Solutions to Graph of

Two unequal real solutions

If a, b, and c are rational numbers and the discriminant is a perfect square, the solutions are rational.

If the discriminant is not a perfect square, the solutions are irrational conjugates.

Two x-intercepts

acb 42

02 cbxax

02 cbxax cbxaxy 2

042 acb

Page 580





One real solution (a repeated solution);

If a, b, and c are rational numbers the repeated solution is also a rational number

One x-intercept

acb 42

02 cbxax

02 cbxax cbxaxy 2

042 acb

CONTINUECONTINUEDD



042 acb

CONTINUECONTINUEDD



No real solution; two imaginary solutions;

The solutions are complex conjugates.

No x-intercepts

acb 42

02 cbxax

02 cbxax cbxaxy 2



EXAMPLEEXAMPLE

For each equation, compute the discriminant. Then determine the number and type of solutions:

SOLUTIONSOLUTION

.25204(b)0342(a) 22 xxxx

a = 2 b = -4 c = 3

Begin by identifying the values for a, b, and c in each equation. Then compute , the discriminant.acb 42

0342(a) 2 xx

Substitute and compute the discriminant:



25204(b) 2 xx

a = 4 b = -20 c = 25

.8241632444 22 acb

We must first put the quadratic equation in standard form.

CONTINUECONTINUEDD

The discriminant, -8, shows that there are two imaginary solutions. These solutions are complex conjugates of each other.

25204 2 xx

025204 2 xx Subtract 20x – 25 from both sides.

Substitute and compute the discriminant:



.04004002544204 22 acb

CONTINUECONTINUEDD

The discriminant, 0, shows that there is only one real solution. This real solution is a rational number.


Quadratic Formula in Application

EXAMPLE: similar to # 79 in homeworkEXAMPLE: similar to # 79 in homework

The hypotenuse of a right triangle is 6 feet long. One leg is 2 feet shorter than the other. Find the lengths of the legs. Round to the nearest tenth of a foot.

SOLUTIONSOLUTION

Since the hypotenuse is 6 feet long, and one leg of the triangle, x, is 2 feet longer than the other leg, x - 2, the triangle can be represented as follows.

x - 2

x

6



Now we can use the Pythagorean Theorem to create an equation that contains the information provided.

CONTINUECONTINUEDD

222 62 xx This is the Pythagorean Theorem.

36222 222 xxx Evaluate the exponents.

36442 2 xx Simplify.

03242 2 xx Subtract 36 from both sides.

01622 2 xx Factor 2 out of all terms on left side.

01622 xx Divide both sides by 2.

Determine a, b, and c to use the quadratic formula.

a = 1 b = -2 c = -16



CONTINUECONTINUEDD

Substitute the values for a, b, and c into the quadratic formula.

12

161422 2

x

Simplify.2

6442 x

Simplify.2

682 x

Rewrite the radicand.2

1742 x

Rewrite as two radicals.2

1742 x



CONTINUECONTINUEDD

Simplify.2

1722 x

Factor 2 out of the numerator. 2

1712 x

Divide the numerator and denominator by 2.

171x

Now and to the nearest tenth of a foot. The answer -3.1 feet is of course impossible. Therefore, the length of the side labeled x must be 5.1 feet. Therefore, the side labeled x – 2 must be 5.1 – 2 = 3.1 feet.

feet 1.5171 x feet 1.3171 x



Check 6 on page 585Check 6 on page 585

The function

Models a woman’s normal systolic blood pressure, P(A), at age A. Use this function to find the age to the nearest year, of a woman whose normal systolic blood pressure is 115 mm Hg.

10705.001.0)( 2 AAAP

Determine a, b, and c to use the quadratic formula.



CONTINUECONTINUEDD

Substitute the values for a, b, and c into the quadratic formula.

Simplify.



CONTINUECONTINUEDD

Simplify.



Problem 84 on page 587: similar to #83 in Problem 84 on page 587: similar to #83 in homeworkhomework



Problem 84 on page 587, continuedProblem 84 on page 587, continued


In Conclusion – an error to watch for

Note:

Many students use the quadratic formula correctly until the last step, where they make an error in simplifying the solutions. Be sure that you factor the numerator before dividing the numerator and denominator by the greatest common factor. Remember. you can only cancel factors of the whole numerator. You cannot divide just one term in the numerator and denominator by their greatest common factor. See page 578 in your text for astudy tip on this common error that many students make.



We can use the method of completing the square to derive an equation that can be used to solve any quadratic equation – those that factor, and those that don’t.

This equation will enable you to solve equations more quickly than the method of completing the square. When quadratics are easy to factor, you will probably want to continue to use the method of factoring, for that will be quicker.

The formula that we will derive and use is called the quadratic formula. You will want to memorize this formula.


Solving Quadratic Equations

Determining the Most Efficient Technique to Use When Solving a Quadratic Equation

Description and Form of the Quadratic Equation

Most Efficient Solution Method

Example

and

can be factored easily.

Factor and use the zero-product principle.

The quadratic equation has no x-term. (b = 0)

Solve for and apply the square root property.

2x

02 cbxax

cbxax 2

02 cax

0253 2 xx

0213 xx

02or 013 xx

2 3

1 xx

074 2 x74 2 x

4/72 x

2/72 x






Example

; u is a first-degree polynomial.

Use the square root property.

du 2 54 2 x

CONTINUECONTINUEDD

54 x

54 x






Example

and

cannot be factored or the factoring is too difficult.

Use the quadratic formula.

0622 xx

CONTINUECONTINUEDD

02 cbxaxcbxax 2

a = 1 b = -2 c = -6

12

61422 2

x

71x


The Zero-Product Principle

The Zero-Product Principle in ReverseIf A = 0 or B = 0, then AB = 0.



EXAMPLEEXAMPLE

Write a quadratic equation with the given solution set:

SOLUTIONSOLUTION

.3

1,

6

5

Because the solution set is , then

3

1,

6

5

3

1 or

6

5 xx

Obtain zero on one side of each equation.

06

5x 0

3

1x

Clear fractions, multiplying by 6 and 3 respectively.

056 x 013 x



Use the zero-product principle in reverse.

01356 xx

CONTINUECONTINUEDD

Use the FOIL method to multiply.0515618 2 xxx

Combine like terms.05918 2 xx

Thus, one equation is . Many other quadratic

equations have for their solution sets.

05918 2 xx

3

1,

6

5

§ 8.2 the quadratic formula. blitzer, intermediate algebra, 5e – slide #2 section 8.2 the...

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