SECTION 6.6 Applications of Quadratic Equations^401
Step 3 Write an equation.The area of a right triangle is given by the formula
In a right triangle, the legs are the base and height, so we substitute 24 for
the area, xfor the base, and for the height in the formula.
Formula for the area of a triangleLetStep 4 Solve. Multiply by 2.
Distributive property
Standard form
Factor.
Zero-factor property
Solve each equation.Step 5 State the answer.The solutions are and 8. Because a triangle cannot
have a side of negative length, we discard the solution Then the lengths
of the legs will be 8 m and
Step 6 Check.The length of one leg is 2 m less than the length of the other leg, and
the area is
as required. NOW TRY
1
2
182162 =24 m^2 ,
8 - 2 = 6 m.
- 6.
- 6
x=- 6 or x= 8
x+ 6 = 0 or x- 8 = 0
1 x+ 621 x- 82 = 0
x^2 - 2 x- 48 = 0
48 =x^2 - 2 x
48 =x 1 x- 22
24 = a=24, b=x, h=x-2.
1
2
x 1 x- 22
a=
1
2
bh
x- 2
area=
1
2
baseheight.
NOW TRY
EXERCISE 1
A right triangle has one leg
that is 4 ft shorter than the
other leg. The area of the tri-
angle is. Determine the
lengths of the legs.
6 ft^2CAUTION In solving applied problems, always check solutions against phys-
ical factsand discard any answers that are not appropriate.
OBJECTIVE 2 Solve problems involving consecutive integers.Recall from
our work in Section 2.4that consecutive integersare integers that are next to each
other on a number line, such as 3 and 4, or and See FIGURE 2(a).
Consecutive odd integersare oddintegers that are next to each other, such as
3 and 5, or and Consecutive even integersare defined similarly—for
example, 4 and 6 are consecutive even integers, as are - 10 and -8.See FIGURE 2(b).
- 13 - 11.
(^0156) - 11 - 10.
x x 1 x 2
234
Consecutive integers
0 1 5 6
x x 2 x 4
2 3 4
Consecutive odd integers
Consecutive even integers
x x 2 x 4
(b)
FIGURE 2
(a)
If xrepresents the lesser integer, then, for any
two consecutive integers, use
three consecutive integers, use
two consecutive even or odd integers, use
three consecutive even or odd integers, use x, x2, x4.
x, x2;
x, x1, x2;
x, x1;
PROBLEM-SOLVING HINT
NOW TRY ANSWER
- 2 ft, 6 ft