Algebra Examples

$y = x^{2} - 6 x + 3$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

$x^{2} - 6 x + 3$ .

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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

$c$ .

$a = 1$

$b = - 6$

$c = 3$

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

$d = \frac{b}{2 a}$ .

$d = \frac{- 6}{2 \cdot 1}$

Step 1.1.3.2

Cancel the common factor of

$- 6$ and

$2$ .

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Step 1.1.3.2.1

Factor

$2$ out of

$- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

$2$ out of

$2 \cdot 1$ .

$d = \frac{2 \cdot - 3}{2 (1)}$

Step 1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 1.1.3.2.2.3

Rewrite the expression.

$d = \frac{- 3}{1}$

Step 1.1.3.2.2.4

Divide

$- 3$ by

$1$ .

$d = - 3$

$d = - 3$

$d = - 3$

$d = - 3$

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

$e = c - \frac{b^{2}}{4 a}$ .

$e = 3 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

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

$- 6$ to the power of

$2$ .

$e = 3 - \frac{36}{4 \cdot 1}$

Step 1.1.4.2.1.2

Multiply

$4$ by

$1$ .

$e = 3 - \frac{36}{4}$

Step 1.1.4.2.1.3

Divide

$36$ by

$4$ .

$e = 3 - 1 \cdot 9$

Step 1.1.4.2.1.4

Multiply

$- 1$ by

$9$ .

$e = 3 - 9$

$e = 3 - 9$

Step 1.1.4.2.2

Subtract

$9$ from

$3$ .

$e = - 6$

$e = - 6$

$e = - 6$

Step 1.1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - 3)}^{2} - 6$ .

${(x - 3)}^{2} - 6$

${(x - 3)}^{2} - 6$

Step 1.2

Set

$y$ equal to the new right side.

$y = {(x - 3)}^{2} - 6$

$y = {(x - 3)}^{2} - 6$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = 1$

$h = 3$

$k = - 6$

Step 3

Find the vertex

$(h, k)$ .

$(3, - 6)$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$