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Intermediate Algebra

OPENSTAX

$Math Studyset Vaia Explanations$ Math

OER 2017 Edition · 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 231 (page 1148)

The sum of the squares of two numbers is 113. The difference of the number is 1. Find the numbers.

Short Answer

Expert verified

-7 and -8 or 8 and 7

Step by step solution

01

Step 1. Given information.

The sum of the squares of two numbers is 113 and the difference between the two numbers is 1.

02

Step 2. Forming the equations.

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

From the condition that the sum of the squares of the two numbers is 113, we get the following equation:

$x^{2} + y^{2} = 113 - - - > (1)$

From the condition that the difference of the two numbers is 1, we get the following equation:

$x - y = 1 - - - > (2)$

03

Step 3. Solving the equations.

Solving equation (2) for $x$ gives

$x = y + 1$

Substituting this in equation (1) gives

${(y + 1)}^{2} + y^{2} = 113 y^{2} + 2 y + 1 + y^{2} = 113 2 y^{2} + 2 y - 112 = 0 - - - > (3)$

Dividing both sides by 2 gives

$y^{2} + y - 56 = 0$

Factorising the quadratic equation gives

$y^{2} + 8 y - 7 y - 56 = 0 (y + 8) (y - 7) = 0$

If $y + 8 = 0$ , then $y = - 8$ and $x = 1 + y = 1 + (- 8) = - 7$

If $y - 7 = 0$ , then $y = 7$ and $x = 1 + y = 1 + 7 = 8$

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