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# If a+b=−8, and a=20/b, what is the value of a^2+b^2?

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If a+b=−8, and a=20/b, what is the value of a^2+b^2?  [#permalink]

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25 Aug 2015, 12:34
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Difficulty:

35% (medium)

Question Stats:

65% (01:39) correct 35% (02:41) wrong based on 127 sessions

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If a+b=−8, and a=20/b, what is the value of a^2+b^2?

A. 24
B. 32
C. 34
D. 40
E. 104

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Re: If a+b=−8, and a=20/b, what is the value of a^2+b^2?  [#permalink]

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25 Aug 2015, 13:12
1
This one threw me off as my original method dealt with complex numbers and even though it is solvable that way I knew there had to be a more efficient way:

a^2 + b^2 should make you think of these formulas:
(a+b)(a+b) = a^2+b^2+2ab
(a+b)(a-b) = a^2-b^2
(a-b)(a-b) = a^2+b^2 - 2ab

The first formula is the key here. We already know (a+b) = -8 and a*b = 20
(a+b)(a+b) = (-8)(-8) = a^2+b^2+2*(20)
a^2+b^2 = 64 - 40 = 24.

Honestly, this question will sure mess up a lot of people who can't see this shortcut, because everyone will say "hell no this can't be right" if they start getting complex numbers. =)
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Re: If a+b=−8, and a=20/b, what is the value of a^2+b^2?  [#permalink]

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25 Aug 2015, 18:18
2
Sum of squares is the most efficient way to solve the problem.

Given: $$a^{2}+b^{2}$$
Solve via sum of squares:
$$(a+b)^{2}-2ab$$
$$(-8)^{2}-2(20) = 24$$

A.
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Re: If a+b=−8, and a=20/b, what is the value of a^2+b^2?  [#permalink]

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20 Nov 2017, 10:17
Top Contributor
reto wrote:
If a+b=−8, and a=20/b, what is the value of a^2+b^2?

A. 24
B. 32
C. 34
D. 40
E. 104

Given: a + b = −8 and a = 20/b

Take a = 20/b and multiply both sides by b to get: ab = 20

Take a + b = −8 and SQUARE both sides to get (a+b)² = (−8)²
Expand and simplify left side to get: a² + 2ab + b² = 64
Now replace ab with 20 to get: a² + 2(20)+ b² = 64
Simplify: a² + 40 + b² = 64
Subtract 40 from both sides to get: a² + b² = 24

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Re: If a+b=−8, and a=20/b, what is the value of a^2+b^2?  [#permalink]

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22 Nov 2017, 13:17
1
reto wrote:
If a+b=−8, and a=20/b, what is the value of a^2+b^2?

A. 24
B. 32
C. 34
D. 40
E. 104

We can square the first equation and we have:

a^2 + b^2 + 2ab = 64

Simplifying the second equation, we have:

ab = 20

Thus:

a^2 + b^2 + 40 = 64

a^2 + b^2 = 24

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Re: If a+b=−8, and a=20/b, what is the value of a^2+b^2? &nbs [#permalink] 22 Nov 2017, 13:17
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