Question

Write y = x2 −14x + 52 in vertex form.


A) y = (x + 7)2 + 3


B) y = (x − 7)2 − 3


C) y = (x − 7)2 + 3


D) y = (x + 7)2 − 3

Answer 1

Option c

Answer:

The vertex form for $y = x^{2} - 14 x + 52 \text { is }\bold{(x - 7)^{2} + 3}$              

Solution:  

The standard form of equations is given as $\bold{y = a x^{2} + bx + c}$

The general representation of vertex form is $\bold{y = a(x-h)^{2} + k}$ where (h,k) is the vertex of the parabola.

The "a" in the vertex form is the same "a" as in $y = a x^{2} + bx + c$ (standard form)

The value of “a” is same in both standard and vertex form.

In order to represent the given expression into vertex form follow the below steps:

From question, Given equation is$y = x^{2} - 14x + 52$.This equation can be rewritten as

$y = x^{2} - 14x + 49 + 3$

Where “52” has been rewritten as 49 + 3.

Now, 14x can be written as 2(7x) and 49 can be written as $7^{2}$. Hence the above equation becomes,

$y = x^{2} - 2(7 x) + 7^{2} + 3$

The first three terms of above equation is of the form $a^{2} - 2ab + b^{2}$, where a = 1 and b = 7

We know that $a^{2} - 2ab + b^{2} = (a - b)^{2}$  

Hence $y = x^{2} - 2(7x) + 7^{2} + 3$ becomes,

$x^{2} - 2(7 x) + 7^{2} + 3 = (x - 7)^{2} + 3$

Now$y = (x - 7)^{2} + 3$ is of the form $y = a(x - h)^{2} + k$

Where by comparing we get, a = 1 and h = 7 and k = 3

$y = (x - 7)^{2} + 3$

Hence the vertex form of $y = x^{2} - 14x + 52 \text { is }\bold{(x - 7)^{2} + 3}$