Write the quadratic function f(x) = x2 + 8x + 3 in vertex form? A) f(x) = (x - 4)2 - 13 B) f(x) = (x - 4)2 + 3 C) f(x) = (x + 4)2 + 3 D) f(x) = (x + 4)2 - 13

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Write the quadratic function f(x) = x2 + 8x + 3 in vertex form? A) f(x) = (x - 4)2 - 13 B) f(x) = (x - 4)2 + 3 C) f(x) = (x + 4)2 + 3 D) f(x) = (x + 4)2 - 13

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Answer 1 · 2016-03-27T01:11:25

$D : f (x) = {(x + 4)}^{2} - 13$ $"D": f(x)=(x+4)^2-13$

Explanation:

Given the following function, you are asked to convert it to vertex form:

$f (x) = x^{2} + 8 x + 3$ $f(x)=x^2+8x+3$

The given possible solutions are:

$A) f (x) = {(x - 4)}^{2} - 13$ $"A") f(x)=(x-4)^2-13$
$B) f (x) = {(x - 4)}^{2} + 3$ $"B") f(x)=(x-4)^2+3$
$C) f (x) = {(x + 4)}^{2} + 3$ $"C") f(x)=(x+4)^2+3$
$D) f (x) = {(x + 4)}^{2} - 13$ $"D") f(x)=(x+4)^2-13$

Converting to Vertex Form
$1$ $1$ . Start by placing brackets around the first two terms.

$f (x) = x^{2} + 8 x + 3$ $f(x)=x^2+8x+3$

$f (x) = (x^{2} + 8 x) + 3$ $f(x)=(x^2+8x)+3$

$2$ $2$ . In order to make the bracketed terms a perfect square trinomial, we must add a " $c$ $color(darkorange)c$ " term as in $a x^{2} + b x + c$ $ax^2+bx+color(darkorange)c$ . Since $c$ $color(darkorange)c$ , in a perfect square trinomial is denoted by the formula $c = {(\frac{b}{2})}^{2}$ $color(darkorange)c=(color(blue)b/2)^2$ , take the value of $b$ $color(blue)b$ to find the value of $c$ $color(darkorange)c$ .

$f (x) = (x^{2} + 8 x + {(\frac{8}{2})}^{2}) + 3$ $f(x)=(x^2+color(blue)8x+(color(blue)8/2)^2)+3$

$3$ $3$ . However, adding ${(\frac{8}{2})}^{2}$ $(8/2)^2$ would change the value of the equation. Thus, subtract ${(\frac{8}{2})}^{2}$ $(8/2)^2$ from the ${(\frac{8}{2})}^{2}$ $(8/2)^2$ you just added.

$f (x) = (x^{2} + 8 x + {(\frac{8}{2})}^{2} - {(\frac{8}{2})}^{2}) + 3$ $f(x)=(x^2+8x+(8/2)^2-(8/2)^2)+3$

$4$ $4$ . Multiply $(- {(\frac{8}{2})}^{2})$ $(-(8/2)^2)$ by the $a$ $color(violet)a$ term as in $a x^{2} + b x + c$ $color(violet)ax^2+bx+c$ to bring it outside the brackets.

$f (x) = (1 x^{2} + 8 x + {(\frac{8}{2})}^{2}) + 3 - ({(\frac{8}{2})}^{2} \times 1)$ $f(x)=(color(violet)1x^2+8x+(8/2)^2)+3-((8/2)^2xxcolor(violet)1)$

$5$ $5$ . Simplify.

$f (x) = (x^{2} + 8 x + 16) + 3 - 16$ $f(x)=(x^2+8x+16)+3-16$

$f (x) = (x^{2} + 8 x + 16) - 13$ $f(x)=(x^2+8x+16)-13$

$6$ $6$ . Lastly, factor the perfect square trinomial.

$color(green)(|bar(ul(color(white)(a/a)f(x)=(x+4)^2-13color(white)(a/a)|)))$

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Write the quadratic function f(x) = x2 + 8x + 3 in vertex form? A) f(x) = (x - 4)2 - 13 B) f(x) = (x - 4)2 + 3 C) f(x) = (x + 4)2 + 3 D) f(x) = (x + 4)2 - 13

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