Chapter 11: Q. 71 (page 774)

The difference of two numbers is $2$ , and the sum of their
squares is $10$ . Find the numbers.

Short Answer

Expert verified

The numbers are $3$ and $1$ or $- 1$ and $- 3$ .

Step by step solution

Step 1. Given information

Given that,

The difference of the two number is $2$ . So, $a - b = 2$ .

The sum of their squares is $10$ . So, $a^{2} + b^{2} = 10$ .

So, the system of equations are,

$\{\begin{cases} a - b = 2 \\ a^{2} + b^{2} = 10 \end{cases}$

Step 2 Let us write the first equation as,

$a - b = 2 a = b + 2$

Substitute the value of $a$ in the first equation.

${(b + 2)}^{2} + b^{2} = 10 b^{2} + 4 b + 4 + b^{2} = 10 2 b^{2} + 4 b + 4 - 10 = 0 2 b^{2} + 4 b - 6 = 0$

Now factor out the common terms.

$2 (b^{2} + 2 b - 3) = 0 b^{2} + 2 b - 3 = 0$

Step 3 we can solve for b by using the quadratic formula -b±b2-4ac2a where a=1,b=2,c=-3.

$b = \frac{- 2 \pm \sqrt{2^{2} - 4 (1) (- 3)}}{2 (1)} b = \frac{- 2 \pm 4}{2} b = \frac{- 2 + 4}{2} o r b = \frac{- 2 - 4}{2} b = \frac{2}{2} o r b = \frac{- 6}{2} b = 1 o r b = - 3$

Step 4 Substitute the value of b in the first equation.

Put $b = 1$ in the first equation.

$a - 1 = 2 a = 2 + 1 a = 3$

Put $b = - 3$ in the first equation.

$a - (- 3) = 2 a + 3 = 2 a = 2 - 3 a = - 1$

So, the required numbers are, $3$ and $1$ or $- 1$ and $- 3$ .

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The difference of two numbers is $2$ , and the sum of their
squares is $10$ . Find the numbers.

Short Answer

Step by step solution

Step 1. Given information

Step 2 Let us write the first equation as,

Step 3 we can solve for b by using the quadratic formula -b±b2-4ac2a where a=1,b=2,c=-3.

Step 4 Substitute the value of b in the first equation.

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The difference of two numbers is 2, and the sum of theirsquares is 10. Find the numbers.

Short Answer

Step by step solution

Step 1. Given information

Step 2 Let us write the first equation as,

Step 3 we can solve for b by using the quadratic formula -b±b2-4ac2a where a=1,b=2,c=-3.

Step 4 Substitute the value of b in the first equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Applied Mathematics

Discrete Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

The difference of two numbers is $2$ , and the sum of their
squares is $10$ . Find the numbers.