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If y= (x-1)(x+2), then what is the least possible value of y

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If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 07 Jul 2013, 06:39
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If y= (x-1)(x+2), then what is the least possible value of y?

A. -3
B. -9/4
C. -2
D. -3/2
E. 0

Any alternative solutions?
[Reveal] Spoiler: OA

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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 07 Jul 2013, 06:55
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fozzzy wrote:
If y= (x-1)(x+2), then what is the least possible value of y?

A. -3
B. -9/4
C. -2
D. -3/2
E. 0

Any alternative solutions?


The function is a parabola with a positive "a" coefficient

The roots are x=1 and x=-2, for values between 1 and -2 it will have negative values (E is out)

\(y=x^2+x-2\), you can try to insert values (starting from the least) to see if it could be the answer. Example
\(-3=x^2+x-2\) or \(x^2+x+1=0\) =>Impossible
\(-\frac{9}{4}= x^2+x-2\) or \(x^2+x+\frac{1}{4}=0\) => \(x=-0.5\) valid => CORRECT

Or approach #2:

Given the roots -2 and 1, the x of the vertex will be the middle point => \(x=-0.5\)

And the least value will be the y coordinate of the vertex (plug \(x=-0.5\) into the equation).
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 07 Jul 2013, 06:58
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fozzzy wrote:
If y= (x-1)(x+2), then what is the least possible value of y?

A. -3
B. -9/4
C. -2
D. -3/2
E. 0

Any alternative solutions?


\(y= (x-1)(x+2)=x^2+x-2\).

Theory:
Quadratic expression \(ax^2+bx+c\) reaches its extreme values when \(x=-\frac{b}{2a}\).
When \(a>0\) extreme value is minimum value of \(ax^2+bx+c\) (maximum value is not limited).
When \(a<0\) extreme value is maximum value of \(ax^2+bx+c\) (minimum value is not limited).

You can look at this geometrically: \(y=ax^2+bx+c\) when graphed on XY plane gives parabola. When \(a>0\), the parabola opens upward and minimum value of \(ax^2+bx+c\) is y-coordinate of vertex, when \(a<0\), the parabola opens downward and maximum value of \(ax^2+bx+c\) is y-coordinate of vertex.
Image

Examples:
Expression \(5x^2-10x+20\) reaches its minimum when \(x=-\frac{b}{2a}=-\frac{-10}{2*5}=1\), so minimum value is \(5x^2-10x+20=5*1^2-10*1+20=15\).

Expression \(-5x^2-10x+20\) reaches its maximum when \(x=-\frac{b}{2a}=-\frac{-10}{2*(-5)}=-1\), so maximum value is \(-5x^2-10x+20=-5*(-1)^2-10*(-1)+20=25\).

Back to the original question:
\(y= (x-1)(x+2)=x^2+x-2\) --> y reaches its minimum (as \(a=1>0\)) when \(x=-\frac{b}{2a}=-\frac{1}{2}\).

Therefore \(y_{min}=(-\frac{1}{2}-1)(-\frac{1}{2}+2)=-\frac{9}{4}\)


Answer: B.

Or:

Use derivative: \(y'=2x+1\) --> equate to 0: \(2x+1=0\) --> \(x=-\frac{1}{2}\) --> \(y_{min}=(-\frac{1}{2}-1)(-\frac{1}{2}+2)=-\frac{9}{4}\)

Answer: B.

Hope it's clear.
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 07 Jul 2013, 07:04
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Here's how I did this one...

\(x^2 + x - 2\)

we can complete the square since \((x+a)^2 = x^2 + 2ax + a^2\)

Here \(2ax = x\) >>\(a=\frac{1}{2}\)

\((x+1/2)^2 - 1/4 - 2 = y\) subtracting 1/4 since 1/4 will be added when its squared.

\((x+1/2)^2 - 9/4 = y\)

\(y = -9/4\)
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 07 Jul 2013, 09:41
Expert's post
fozzzy wrote:
If y= (x-1)(x+2), then what is the least possible value of y?

A. -3
B. -9/4
C. -2
D. -3/2
E. 0

Any alternative solutions?


\(y = x^2+x-2 \to x^2+x-(2+y)=0 \to\) For real values of x, the Discriminant\((D)\geq{0} \to 1^2-4*1*[-(2+y)]\geq{0}\to\)

\(y+2\geq{-\frac{1}{4} }\to y\geq{-\frac{9}{4}}\)

B.
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 27 Oct 2014, 21:04
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 28 Oct 2014, 12:08
Can we also use b^2 - 4ac to determine the valid minimum value, as thats what I did.. I got 0 as the answer for B and I thought 0 means invalid so went for C. I want some clarification here, first if b^2-4ac=0, it means eq has no real soln?? Or my approach for this question was wrong ! Thanks in advance :)
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 29 Oct 2014, 01:09
I did it this way
The minimum (or max, if the coef is -a) should lie half way between the two roots which are -2 and 1. halfway between that is -3/2.
Now put in -3/2 in place of x:
(-3/2 -1)(-3/2 +2) = -9/4
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 29 Oct 2014, 02:40
can someone tell me why it is wrong to just test all the answer values for for x and find the lowest number? That would get you to answer E (-2).

I missed a question once because I assumed that a y= (x equation). Meant x equation was equal to zero. When do we assume it's a quadratic equation?
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Re: If y= (x-1)(x+2), then what is the least possible value of y [#permalink] New post 29 Oct 2014, 12:57
When X has a power 2. Quad comes from the word "square"
Re: If y= (x-1)(x+2), then what is the least possible value of y   [#permalink] 29 Oct 2014, 12:57
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