Algebra Examples

$y = 4 x^{2} - 12 x + 9$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $4 x^{2} - 12 x + 9$ .

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Step 1.1.1

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

$a = 4$

$b = - 12$

$c = 9$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

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

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Step 1.1.3.1

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

$d = \frac{- 12}{2 \cdot 4}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $- 12$ and $2$ .

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Step 1.1.3.2.1.1

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 4}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 6}{2 (4)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 4}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 6}{4}$

Step 1.1.3.2.2

Cancel the common factor of $- 6$ and $4$ .

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Step 1.1.3.2.2.1

Factor $2$ out of $- 6$ .

$d = \frac{2 (- 3)}{4}$

Step 1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.3.2.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot - 3}{2 \cdot 2}$

Step 1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 2}$

Step 1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{- 3}{2}$

Step 1.1.3.2.3

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

Step 1.1.4

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

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Step 1.1.4.1

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

$e = 9 - \frac{{(- 12)}^{2}}{4 \cdot 4}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise $- 12$ to the power of $2$ .

$e = 9 - \frac{144}{4 \cdot 4}$

Step 1.1.4.2.1.2

Multiply $4$ by $4$ .

$e = 9 - \frac{144}{16}$

Step 1.1.4.2.1.3

Divide $144$ by $16$ .

$e = 9 - 1 \cdot 9$

Step 1.1.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 9 - 9$

Step 1.1.4.2.2

Subtract $9$ from $9$ .

$e = 0$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - \frac{3}{2})}^{2} + 0$ .

$4 {(x - \frac{3}{2})}^{2} + 0$

Step 1.2

Set $y$ equal to the new right side.

$y = 4 {(x - \frac{3}{2})}^{2} + 0$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 4$

$h = \frac{3}{2}$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(\frac{3}{2}, 0)$

Step 4