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Find the minimum value of x^2 + 7x + 14.

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Find the minimum value of x^2 + 7x + 14. [#permalink] New post   12 Jul 2019, 00:53
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Find the minimum value of \(x^2 + 7x + 14\).

A. 14
B. -7
C. 0
D. 7
E. 1.75
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E
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Bunuel
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   12 Jul 2019, 01:06
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Dillesh4096 wrote:
Find the minimum value of \(x^2 + 7x + 14\).

A. 14
B. -7
C. 0
D. 7
E. 1.75


\(x^2 + 7x + 14=x^2+2*3.5x + 3.5^2 + 1.75=(x + 3.5)^2 + 1.75\).

The least value of (x + 3.5)^2 is obviously 0 (for x = -3.5), thus the least value of the whole expression is 1.75.

Answer: E.
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sukoon9334
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   16 Oct 2023, 08:06
How do we find roots of such questions?
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   16 Oct 2023, 09:07
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sukoon9334 wrote:
How do we find roots of such questions?


Although finding the roots of this expression isn't necessary to answer the question, as shown by Bunuel, if you are still interested, you can determine them using the below formula:
\(x= \frac{−b±\sqrt{b^2-4ac}}{2a} \)
where a=coefficient of x^2, b=coefficient of x and c=constant

So, for this expression, the roots will be:
​\(x= \frac{−7±\sqrt{49-4*1*14}}{2*1} = \frac{−7±\sqrt{-7}}{2} \)
Since the roots contain square root of a negative number, the roots will be complex numbers as follows:
\(x=\frac{-7}{2}+\frac{\sqrt{7}}{2}i\) and \(x=\frac{-7}{2}-\frac{\sqrt{7}}{2}i\)

Hope this helps!
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   11 Dec 2023, 18:37
Using differential equations to find the maxima/minima is a huge time saver on GMAT (if you can get comfortable with it). Here's a great link/resource. It is super simple in my opinion.

https://www.mathsisfun.com/calculus/maxima-minima.html
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   11 Dec 2023, 22:57
Calcuakte the differential of equation w.r.t x, which comes out 2x+7. Equate this with zero and x = -7/2. Put this value in the orginal equation to calculate minimum value that comes out 1.75.

For equation with no complex roots, maxima/minma concept of differential equation is used.

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Re: Find the minimum value of x^2 + 7x + 14. [#permalink] New post   13 Dec 2023, 12:18
Use the formula x=-(b/2a) to calculate maxima or minima given coefficient a>0 or <0.
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Re: Find the minimum value of x^2 + 7x + 14. [#permalink]
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