Download 368 14–1 Solving Quadratics by Factoring General Form of a

MATH AROUND US
BEING RESOURCEFUL IN HUMAN RESOURCES—ELYSE PAYNE
Elyse Payne works as a human resources generalist, supporting employees at a Crown
corporation. Though she didn’t expect it, using math is a daily part of her job.
Each day Elyse finds herself doing basic math like adding, subtracting, and calculating
percentages. However, for her, the most surprising use of her math skills has come from her job’s
heavy use of Microsoft Excel®. She says, “I use the Formula bar in Excel daily. It calls upon my
math skills because I have to think of how the calculation will come out before I make entries or
create functions.”
While Elyse did well in her math courses throughout high school, university, and college, it wasn’t
clear how the subject could eventually apply to her future career aspects. However, now that
she is out of school and in the workforce, she believes that education helps contribute to her
success. “It’s because of my math skills that I can use an application like Excel® easily and
successfully, making me a more productive and valuable employee to my employer.”
14–1
Recall that in a polynomial, all
powers of x are positive integers.
Solving Quadratics by Factoring
Terminology
A polynomial equation of second degree is called a quadratic equation. It is common practice
to refer to it simply as a quadratic.
◆◆◆
Example 1: The following equations are quadratic equations:
(a) 4x2 5x 2 0
(b) x2 58
(c) 9x2 5x 0
(d) 2x2 7 0
Equation (a) is called a complete quadratic; (b) and (d), which have no x terms, are called pure
◆◆◆
quadratics; and (c), which has no constant term, is called an incomplete quadratic.
A quadratic is in general form when it is written in the following form, where a, b, and c
are constants:
General
Form of a
Quadratic
◆◆◆
ax2 bx c 0
99
Example 2: Write the quadratic equation
5x2
7 4x 3
in general form, and identify a, b, and c.
Solution: Subtracting 5x2/3 from both sides and writing the terms in descending order of the
exponents, we obtain
5x2
4x 7 0
3
Quadratics in general form are usually written without fractions and with the first term positive.
Multiplying by 3, we get
5x2 12x 21 0
368
Section 14–1
◆
369
Solving Quadratics by Factoring
The equation is now in general form, with a 5, b 12, and c 21.
◆◆◆
Number of Roots
We’ll see later in this chapter that the maximum number of solutions that certain types of equations may have is equal to the degree of the equation. Thus a quadratic equation, being of degree
2, has two solutions or roots. The two roots are sometimes equal, or they may be imaginary or
complex numbers. However, in applications, we’ll see that one of the two roots must sometimes
be discarded.
◆◆◆
Example 3: The quadratic equation
x2 4
has two roots, x 2 and x 2.
◆◆◆
Solving Pure Quadratics
To solve a pure quadratic, we simply isolate the x2 term and then take the square root of both
sides, as in the following example.
◆◆◆
Example 4: Solve 3x2 75 0.
Solution: Adding 75 to both sides and dividing by 3, we obtain
3x2 75
x2 25
Taking the square root yields
x 25 5
Check: We check our solution the same way as for other equations, by substituting back into
the original equation. Now, however, we must check two solutions.
Substitute 5:
3(5)2 75 0
75 75 0 (checks)
Substitute 5: 3(5)2 75 0
75 75 0 (checks)
◆◆◆
When taking the square root, be sure to keep both the plus and the minus
values. Both will satisfy the equation.
At this point students usually grumble: “First we’re told that 4 2 only, not 2
(Sec. 1–5). Now we’re told to keep both plus and minus. What’s going on here?” Here’s the
difference: When we solve a quadratic, we know that it must have two roots, so we keep both
values. When we evaluate a square root, such as 4 , which is not the solution of a quadratic,
we agree to take only the positive value to avoid ambiguity.
The roots might sometimes be irrational, as in the following example.
◆◆◆
Example 5: Solve 4x2 15 0.
Solution: Following the same steps as before, we get
4x2 15
15
x2 4
15
15
x 4
2
◆◆◆
370
Chapter 14
◆
Quadratic Equations
Solving Incomplete Quadratics
To solve an incomplete quadratic, remove the common factor x from each term, and set each
factor equal to zero.
◆◆◆
Example 6: Solve
x2 5x 0
Solution: Factoring yields
x(x 5) 0
We use this idea often in this
chapter. If we have the product of
two quantities a and b set equal
to zero, ab 0, this equation will
be true if a 0 (0 • b 0) or if
b 0 (a • 0 0), or if both are
zero (0 • 0 0).
Note that this expression will be true if either or both of the two factors equal zero. We therefore
set each factor in turn equal to zero.
x0
x50
x 5
The two solutions are thus x 0 and x 5.
◆◆◆
Do not cancel an x from the terms of an incomplete quadratic. That will cause a root to be lost. In the last example,
if we had said
Common
Error
x2 5x
and had divided by x,
x 5
we would have obtained the correct root x 5 but would
have lost the root x 0.
Solving Complete Quadratics
We now consider a quadratic that has all of its terms in place: the complete quadratic.
General
Form
ax2 bx c 0
99
First write the quadratic in general form, as given in Eq. 99. Factor the trinomial (if possible) by the methods of Chapter 8, and set each factor equal to zero.
◆◆◆
Example 7: Solve by factoring:
x2 x 6 0
Solution: Factoring gives
(x 3)(x 2) 0
This equation will be satisfied if either or both of the two factors (x 3) and (x 2) are zero.
We therefore set each factor in turn to equal zero.
x30
x20
so the roots are
x3
and
x 2
◆◆◆
Section 14–1
◆
371
Solving Quadratics by Factoring
When the product of two quantities is zero, as in
(x 3)(x 2) 0
Common
Error
we can set each factor equal to zero, getting x 3 0 and
x 2 0.
But this is valid only when the product is zero. Thus if
(x 3)(x 2) 5
we cannot say that x 3 5 and x 2 5.
Often an equation must first be simplified before factoring.
◆◆◆
Example 8: Solve for x:
x(x 8) 2x(x 1) 9
Solution: Removing parentheses gives
x2 8x 2x2 2x 9
Collecting terms, we get
x2 6x 9 0
Factoring yields
(x 3)(x 3) 0
which gives the double root,
x 3
◆◆◆
Sometimes at first glance an equation will not look like a quadratic. The following example
shows a fractional equation which, after simplification, turns out to be a quadratic.
◆◆◆
Example 9: Solve for x:
3x 1
4x 7
x1
x7
Solution: We start by multiplying both sides by the LCD, (4x 7)(x 7). We get
(3x 1)(x 7) (x 1)(4x 7)
or
3x2 20x 7 4x2 11x 7
Collecting terms gives
x2 9x 14 0
Factoring yields
(x 7)(x 2) 0
so x 7 and x 2.
Writing the Equation When the Roots Are Known
Given the roots, we simply reverse the process to find the equation.
◆◆◆
372
Chapter 14
◆◆◆
◆
Quadratic Equations
Example 10: Write the quadratic equation that has the roots x 2 and x 5.
Solution: If the roots are 2 and 5, we know that the factors of the equation must be (x 2)
and (x 5). So
(x 2)(x 5) 0
Multiplying gives us
x2 5x 2x 10 0
So
x2 3x 10 0
◆◆◆
is the original equation.
Solving Radical Equations
In Chapter 13 we solved simple radical equations. We isolated a radical on one side of the
equation and then squared both sides. Here we solve equations in which this squaring operation
results in a quadratic equation.
◆◆◆
Example 11: Solve for x:
4
3x 1 4
x 1
Solution: We clear fractions by multiplying through by x 1 .
3(x 1) 4 4 x 1
3x 7 4 x 1
Squaring both sides yields
9x2 42x 49 16(x 1)
Removing parentheses and collecting terms gives
9x2 58x 65 0
Factoring gives
Check: When x 5,
13
When x ,
9
Remember that the squaring
operation sometimes gives an
extraneous root that will not
check.
So
Our solution is then x 5.
(x 5)(9x 13) 0
13
x 5 and x 9
4
ⱨ4
3 5 1 5 1
4
3(2) 4
2
13
9
1 (checks)
9 3
4
2
2 4
3• ⱨ 4
3 2
3
2 6 4 (does not check)
◆◆◆
Section 14–1
Exercise 1
◆
373
Solving Quadratics by Factoring
◆
Solving Quadratics by Factoring
Pure and Incomplete Quadratics
Solve for x.
1. 2x 5x2
2. 2x 40x2 0
3. 3x(x 2) x(x 3)
4. 2x2 6 66
5. 5x2 3 2x2 24
7. (x 2)2 4x 5
9. 8.25x2 2.93x 0
6. 7x2 4 3x2 40
8. 5x2 2 3x2 6
10. 284x 827x2
Complete Quadratics
Solve for x.
11.
13.
15.
17.
x2 2x 15 0
x2 x 20 0
x2 x 2 0
2x2 3x 5 0
12.
14.
16.
18.
x2 6x 16 0
x2 13x 42 0
x2 7x 12 0
4x2 10x 6 0
19. 2x2 5x 12 0
20. 3x2 x 2 0
21. 5x2 14x 3 0
2x
23. 2x2
3
25. x(x 5) 36
22. 5x2 3x 2 0
24. (x 6)(x 6) 5x
5
1
27. x2 x 6
6
7
1
2
28. x4 4x 3
26. (2x 3) 2x x
2
2
Literal Equations
Solve for x.
29. 4x2 16ax 12a2 0
30. 14x2 23ax 3a2 0
31. 9x2 30bx 24b2 0
32. 24x2 17xy 3y2 0
Writing the Equation from the Roots
Write the quadratic equations that have the following roots.
33. x 4 and x 7
3
2
35. x and x 3
5
Radical Equations
Solve each radical equation for x, and check.
37. 5x2 3x 41 3x 7
4
38. 3 x 1 4
x 1
12
39. x 5 x 12
40. 7x 8 5x 4 2
34. x 3 and x 5
36. x p and x q
There are no applications in the
first few exercise sets. These will
come a little later.
374
Chapter 14
14–2
The method of completing the
square is really too cumbersome
to be a practical tool for solving
quadratics. The main reason we
learn it is to derive the quadratic
formula. Furthermore, the
method of completing the square
is a useful technique that we’ll
use again in later chapters.
◆
Quadratic Equations
Solving Quadratics by Completing the Square
If a quadratic equation is not factorable, it is possible to manipulate it into factorable form by a
procedure called completing the square. The form into which we shall put our expression is the
perfect square trinomial, Eqs. 47 and 48, which we studied in Sec. 8–6.
In the perfect square trinomial,
1. The first and last terms are perfect squares.
2. The middle term is twice the product of the square roots of the outer terms.
To complete the square, we manipulate our given expression so that these two conditions
are met. This is best shown by an example.
◆◆◆
Example 12: Solve the quadratic x2 8x 6 0 by completing the square.
Solution: Subtracting 6 from both sides, we obtain
x2 8x 6
We complete the square by adding the square of half the coefficient of the x term to both sides.
The coefficient of x is 8. We take half of 8 and square it, getting (4)2 or 16. Adding 16 to
both sides yields
x2 8x 16 6 16 10
Factoring, we have
(x 4)2 10
Taking the square root of both sides, we obtain
x 4 10
Finally, we add 4 to both sides.
x 4 10
7.16
Common
Error
or
◆◆◆
0.838
When you are adding the quantity needed to complete the
square to the left-hand side, it's easy to forget to add the same
quantity to the right-hand side.
x2 8x 16 6 16
don’t forget
If the x2 term has a coefficient other than 1, divide through by this coefficient before completing the square.
◆◆◆
Example 13: Solve:
2x2 4x 3 0
Solution: Rearranging and dividing by 2 gives
3
x2 2x 2