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If 2 roots of an equation x^2-3x-3=0 are m and n, what is...

by AAPL » Fri Dec 08, 2017 10:18 am
If 2 roots of an equation x^2-3x-3=0 are m and n, what is the value of m^2+n^2?

A. 10
B. 12
C. 15
D. 24
E. 25

The OA is C.

In this PS question I just need to find the roots and then solve the equation that is function of m and n, right? Can I find it using the second grade formula?

Experts, can you give me any sugestion about how can I solve this PS question please? Thanks.
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by GMATGuruNY » Fri Dec 08, 2017 10:29 am
Given x² + bx + c = 0:
-b = the SUM of the two roots.
c = the PRODUCT of the two roots
AAPL wrote:If 2 roots of an equation x^2-3x-3=0 are m and n, what is the value of m^2+n^2?

A. 10
B. 12
C. 15
D. 24
E. 25
In the equation above, b = -3 and c = -3.
Thus:
m+n = -b = -(-3) = 3.
mn = c = -3.

(m+n)² = m² + n² + 2mn.
Substituting m+n = 3 and mn = -3 into (m+n)² = m² + n² + 2mn, we get:
3² = m² + n² + 2(-3)
9 = m² + n² - 6
15 = m² + n².

The correct answer is C.
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by EconomistGMATTutor » Fri Dec 08, 2017 1:34 pm
If 2 roots of an equation x^2-3x-3=0 are m and n, what is the value of m^2+n^2?

A. 10
B. 12
C. 15
D. 24
E. 25

The OA is C.
Hi AAPL,
Let's take a look at your question.

$$x^2-3x-3=0$$
For a quadratic equation, we know that,
Sum of the roots = -b= -(-3) = 3
Product of the roots = c = -3

If roots of the given quadratic equation are m and n, then:
$$m+n=3$$
$$mn=-3$$
Using the polynomial identity:
$$\left(m+n\right)^2=m^2+n^2+2mn$$

Plugin in the known values in above equation.
$$\left(3\right)^2=m^2+n^2+2\left(-3\right)$$
$$9=m^2+n^2-6$$
$$m^2+n^2=9+6$$
$$m^2+n^2=15$$

Therefore, Option C is correct.

Hope it helps.
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by [email protected] » Fri Dec 08, 2017 2:19 pm
Hi AAPL,

We're told that the 2 roots of the equation x^2-3x-3=0 are M and N. We're asked for the value of M^2 + N^2. If you don't immediately recognize the Algebraic patterns involved, you can still use a bit of 'brute force' arithmetic (and the 'spread' of the answer choices) to get to the correct answer.

To start, we know that there are 2 solutions - and the equation is relatively 'simple', so the solutions are likely 'close' to single-digit integers. We can do a few calculations to look for a pattern:

IF...
X = -2... X^2 - 3X - 3 = 7
X = -1... = 1
X = 0... = -3
X = 1... = -5
X = 2... = -5
X = 3... = -3
X = 4... = 1

Looking at this pattern, the 2 solutions will be (between -1 and 0, but closer to -1) and (between 3 and 4, but closer to 4). (-1)^2 + (4)^2 = 17. There's only one answer that's close to that....

Final Answer: C

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