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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 73 (page 774)

The product of two numbers is $4$ , and the sum of their squares is $8$ . Find the numbers.

Short Answer

Expert verified

The two numbers are $2 a n d 2$ or $2 a n d 2$ .

Step by step solution

01

Step 1. Given information.

Assume the two numbers to be xandy.

From the question,

$x 脳 y = 4 . . . . . . (i) x^{2} + y^{2} = 8 . . . . . . (i i)$

Solve equation (i) for x,

$x = \frac{4}{y} . . . . . . (i i i)$

02

Step 2. Substitute values in equation (ii).

Substitute the value of x in equation (ii),

${(\frac{4}{y})}^{2} + y^{2} = 8 \frac{16}{y^{2}} + y^{2} = 8 \frac{16}{y^{2}} + y^{2} - 8 = 0 16 + y^{4} - 8 y^{2} = 0 {(y^{2} - 4)}^{2} = 0$

03

Step 3. Find the two numbers.

Take square root both the sides,

$\sqrt{{(y^{2} - 4)}^{2}} = 0 y^{2} - 4 = 0 y = 卤 2$

Substitute $y = 2$ in equation (iii),

$x = \frac{4}{2} x = 2$

Substitute $y = - 2$ in equation (iii),

$x = \frac{4}{- 2} x = - 2$

Therefore, the numbers are $2 a n d 2$ or $- 2 a n d - 2$ .

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