What is the minimum possible value of x^2 – 6x + 14?

Question

What is the minimum possible value of  x^2 – 6x + 14 ?

A. 0 
B. 1
C. 5
D. 6
E. 14

IanStewart · Accepted Answer

The standard way to solve questions like this (without formulas you'd never need on the GMAT) is by "completing the square". If we first look only at this part of the quadratic:

x^2 - 6x

we can make this into a perfect square by adding 9:

x^2 - 6x + 9 = (x - 3)^2

We see how to do that just by dividing the "-6" by 2. So using that in the original question:

x^2 - 6x + 14 = x^2 - 6x + 9 + 5 = (x - 3)^2 + 5

and since (x-3)^2 is a square, its minimum value is 0 (when x = 3), so the minimum value of (x - 3)^2 + 5 is equal to 5.

I can't recall a single official question where I've needed to use this technique, though.

rimco2552 · Answer

This can be solved by a simple formula:

Max or minimum value of a quadratic equation "ax^2+bx+c" is given by:

max/min = c- (b^2/4a)

Now, if the Coefficient of x^2 (a) is positive, the equation will have a minimum value where else it the same is negative, it will have a maximum value.

Hence, the minimum value of the given eq is:

14 - (36/4) = 14-9 = 5 (Answer : C)

rishab0507 · Answer

For a quadratic function of the form ax^2 + bx + c where a>0, the minimum value of the function is given by  4ac–b^2 /4a
 
Putting values: 
4*1*14 - (6)^2 /4*1 = 20/4 = > 5

Answer C

vaibhav101 · Answer

No formula approach.

x^2 – 6x + 14  =  x^2 – 6x + 8 + 6  =  x^2 – 2x - 4x + 8 + 6  = (x-2) (x-4) + 6

Remember here,  x\geq{2} . Now at x = 2,4 the the value will be 6 , but at x=3, the minimum value will be 5.

So Answer is C.

RajDev95 · Answer

Think of it as 10+x^2. 
For us to get the minimum value, we can only assign x to 0 (x=0). 
Since there is no point in assigning a negative number because of the square.
Now let's work on the question.

x^2–6x+14
x^2-6x+9+5
(x^2-2.x.3+3^2)+5
(x-3)^2+5

So, the minimum value here is 5.

ANS is C.

Satishqwerty · Answer

The minimum value of a quadratic equation occurs at  \frac{-b}{2a}

The minimum value will occur at  \frac{-(-6)}{2 	imes 1} = 3

Putting the value  x = 3  in the equation  x^2 – 6x + 14

3^2 - 6 	imes 3 + 14 = 9 + 14 - 18 = 23 - 18 = 5

OA, C

mSKR · Answer

b^2-4ac>= 0 ( because factor of x can be found by x = -b+(or-)sqrt ( b^2-4ac)/2a
i can only change value of c to find the minimum value of expression ( x^2–6x+14 )
36-4c>0 --> c=<9 ==> c= 9 for expression to have 0 
current value of c is 14 
so minimum value of given expression is 14-9 = 5 ( it would be (x-some factor)^2+ 5= 0

BlueRocketAsh · Answer

x^2–6x+14
= x^2-6x+9+5
= (x-3)^2+5
Minimum value of square term is 0
Therefore minimum value of the expression is 5

bumpbot · Answer

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What is the minimum possible value of x^2 – 6x + 14?

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What is the minimum possible value of x^2 – 6x + 14?

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