Algebra Examples

$y + 3 x - 6 = - 3 {(x - 2)}^{2} + 4$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Subtract $3 x$ from both sides of the equation.

$y - 6 = - 3 {(x - 2)}^{2} + 4 - 3 x$

Step 1.1.2

Add $6$ to both sides of the equation.

$y = - 3 {(x - 2)}^{2} + 4 - 3 x + 6$

Step 1.1.3

Add $4$ and $6$ .

$y = - 3 {(x - 2)}^{2} - 3 x + 10$

Step 1.2

Complete the square for $- 3 {(x - 2)}^{2} - 3 x + 10$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$- 3 ((x - 2) (x - 2)) - 3 x + 10$

Step 1.2.1.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$- 3 (x (x - 2) - 2 (x - 2)) - 3 x + 10$

Step 1.2.1.1.2.2

Apply the distributive property.

$- 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) - 3 x + 10$

Step 1.2.1.1.2.3

Apply the distributive property.

$- 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) - 3 x + 10$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$- 3 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) - 3 x + 10$

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $x$ .

$- 3 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) - 3 x + 10$

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$- 3 (x^{2} - 2 x - 2 x + 4) - 3 x + 10$

Step 1.2.1.1.3.2

Subtract $2 x$ from $- 2 x$ .

$- 3 (x^{2} - 4 x + 4) - 3 x + 10$

Step 1.2.1.1.4

Apply the distributive property.

$- 3 x^{2} - 3 (- 4 x) - 3 \cdot 4 - 3 x + 10$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 4$ by $- 3$ .

$- 3 x^{2} + 12 x - 3 \cdot 4 - 3 x + 10$

Step 1.2.1.1.5.2

Multiply $- 3$ by $4$ .

$- 3 x^{2} + 12 x - 12 - 3 x + 10$

Step 1.2.1.2

Subtract $3 x$ from $12 x$ .

$- 3 x^{2} + 9 x - 12 + 10$

Step 1.2.1.3

Add $- 12$ and $10$ .

$- 3 x^{2} + 9 x - 2$

Step 1.2.2

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

$a = - 3$

$b = 9$

$c = - 2$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{9}{2 \cdot - 3}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $9$ and $- 3$ .

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Step 1.2.4.2.1.1

Factor $3$ out of $9$ .

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

Step 1.2.4.2.1.2

Cancel the common factors.

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Step 1.2.4.2.1.2.1

Factor $3$ out of $2 \cdot - 3$ .

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

Step 1.2.4.2.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2.4.2.3

Move the negative in front of the fraction.

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

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - 2 - \frac{9^{2}}{4 \cdot - 3}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Raise $9$ to the power of $2$ .

$e = - 2 - \frac{81}{4 \cdot - 3}$

Step 1.2.5.2.1.2

Multiply $4$ by $- 3$ .

$e = - 2 - \frac{81}{- 12}$

Step 1.2.5.2.1.3

Cancel the common factor of $81$ and $- 12$ .

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Step 1.2.5.2.1.3.1

Factor $3$ out of $81$ .

$e = - 2 - \frac{3 (27)}{- 12}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $3$ out of $- 12$ .

$e = - 2 - \frac{3 \cdot 27}{3 \cdot - 4}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = - 2 - \frac{3 \cdot 27}{3 \cdot - 4}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = - 2 - \frac{27}{- 4}$

Step 1.2.5.2.1.4

Move the negative in front of the fraction.

$e = - 2 - - \frac{27}{4}$

Step 1.2.5.2.1.5

Multiply $- - \frac{27}{4}$ .

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Step 1.2.5.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = - 2 + 1 (\frac{27}{4})$

Step 1.2.5.2.1.5.2

Multiply $\frac{27}{4}$ by $1$ .

$e = - 2 + \frac{27}{4}$

Step 1.2.5.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = - 2 \cdot \frac{4}{4} + \frac{27}{4}$

Step 1.2.5.2.3

Combine $- 2$ and $\frac{4}{4}$ .

$e = \frac{- 2 \cdot 4}{4} + \frac{27}{4}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 4 + 27}{4}$

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $- 2$ by $4$ .

$e = \frac{- 8 + 27}{4}$

Step 1.2.5.2.5.2

Add $- 8$ and $27$ .

$e = \frac{19}{4}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{3}{2})}^{2} + \frac{19}{4}$ .

$- 3 {(x - \frac{3}{2})}^{2} + \frac{19}{4}$

Step 1.3

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{3}{2})}^{2} + \frac{19}{4}$

Step 2

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

$a = - 3$

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

$k = \frac{19}{4}$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{19}{4})$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 3}$

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{3}{2}, \frac{14}{3})$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

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

Step 8