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Intermediate Algebra

OPENSTAX

$Math Studyset Vaia Explanations$ Math

OER 2017 Edition · 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 230 (page 1148)

The sum of the squares of two numbers is 65. The difference of the number is 3. Find the numbers.

Short Answer

Expert verified

$- 4$ and $- 7$ or 7 and 4

Step by step solution

01

Step 1. Given information.

The sum of the squares of two numbers is 65 and the difference of the numbers is 3.

02

Step 2. Forming the equations.

Let $x$ be the first number and $y$ be the second number.

From the condition that sum of the square of the numbers is 65, we have the following equation:

$x^{2} + y^{2} = 65 - - - > (1)$

From the condition that the difference between the two numbers is 3, we have the following equation:

$x - y = 3 - - - > (2)$

03

Step 3. Solving the equations.

Solving equation (2) for $x$ , we get that

$x = y + 3$

Substituting this in equation (1) gives

${(y + 3)}^{2} + y^{2} = 65$

Simplifying

$y^{2} + 6 y + 9 + y^{2} = 65 2 y^{2} + 6 y = 65 - 9 2 y^{2} + 6 y - 56 = 0$

Dividing both sides by 2

$y^{2} + 3 y - 28 = 0$

Factoring

$y^{2} + 7 y - 4 y - 28 = 0 (y + 7) (y - 4) = 0$

If $y + 7 = 0$ , then $y = - 7$ and $x = 3 + y = 3 + (- 7) = - 4$

If $y - 4 = 0$ , then $y = 4$ and $x = 3 + y = 7$

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Most popular questions from this chapter

Solve: ${(9 x + 9)}^{\frac{1}{4}} - 2 = 1$

In the following exercises, simplify each complex rational expression using either method.

$\frac{\frac{3}{b^{2} - 3 b - 40}}{\frac{5}{b + 5} - \frac{2}{b - 8}}$

In the following exercises, simplify each expression using the properties for exponents.

$ⓐ q^{27} \cdot q^{15} ⓑ 5^{x} \cdot 5^{4 x} ⓒ 9 u^{41} \cdot 7 u^{53} ⓓ c^{5} \cdot c^{11} \cdot c^{2}$

Of all the factoring methods covered in this chapter (GCF, grouping, undo FOIL, 'ac' method, special products) which is the easiest for you? Which is the hardest? Explain your answers.

In the following exercises, multiply the rational expressions. $\frac{5 c^{2} + 9 c + 2}{c^{2} - 25} . \frac{c^{2} + 10 c + 25}{3 c^{2} - 14 c - 5}$

See all solutions

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