Find the Standard Form of the Parabola g(x)=x^2+14x+49

$g (x) = x^{2} + 14 x + 49$

Step 1

Write $g (x) = x^{2} + 14 x + 49$ as an equation.

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

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 3

Subtract $49$ from both sides of the equation.

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

Step 4

Complete the square.

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 14$

$c = 0 = y - 49$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 49$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{14}{2 \cdot 1}$

Cancel the common factor of $14$ and $2$ .

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 7}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{7}{1}$

Divide $7$ by $1$ .

$d = 7$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{14^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $14$ to the power of $2$ .

$e = 0 - \frac{196}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{196}{4}$

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Subtract $49$ from $0$ .

$e = - 49$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 7)}^{2} - 49$ .

${(x + 7)}^{2} - 49 = y - 49$

Step 5

Move all terms not containing $x$ to the right side of the equation.

Add $49$ to both sides of the equation.

${(x + 7)}^{2} = y - 49 + 49$

Combine the opposite terms in $y - 49 + 49$ .

Add $- 49$ and $49$ .

${(x + 7)}^{2} = y + 0$

Add $y$ and $0$ .

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

Find the Standard Form of the Parabola g(x)=x^2+14x+49

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