Algebra Examples

$y = - \frac{2}{3} {(x - 9)}^{2} - 2$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

Rewrite ${(x - 9)}^{2}$ as $(x - 9) (x - 9)$ .

$y = - \frac{2}{3} \cdot ((x - 9) (x - 9)) - 2$

Step 2.2

Expand $(x - 9) (x - 9)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y = - \frac{2}{3} \cdot (x (x - 9) - 9 (x - 9)) - 2$

Step 2.2.2

Apply the distributive property.

$y = - \frac{2}{3} \cdot (x \cdot x + x \cdot - 9 - 9 (x - 9)) - 2$

Step 2.2.3

Apply the distributive property.

$y = - \frac{2}{3} \cdot (x \cdot x + x \cdot - 9 - 9 x - 9 \cdot - 9) - 2$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

$y = - \frac{2}{3} \cdot (x^{2} + x \cdot - 9 - 9 x - 9 \cdot - 9) - 2$

Step 2.3.1.2

Move $- 9$ to the left of $x$ .

$y = - \frac{2}{3} \cdot (x^{2} - 9 \cdot x - 9 x - 9 \cdot - 9) - 2$

Step 2.3.1.3

Multiply $- 9$ by $- 9$ .

$y = - \frac{2}{3} \cdot (x^{2} - 9 x - 9 x + 81) - 2$

Step 2.3.2

Subtract $9 x$ from $- 9 x$ .

$y = - \frac{2}{3} \cdot (x^{2} - 18 x + 81) - 2$

Step 2.4

Apply the distributive property.

$y = - \frac{2}{3} x^{2} - \frac{2}{3} \cdot (- 18 x) - \frac{2}{3} \cdot 81 - 2$

Step 2.5

Simplify.

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Step 2.5.1

Combine $x^{2}$ and $\frac{2}{3}$ .

$y = - \frac{x^{2} \cdot 2}{3} - \frac{2}{3} \cdot (- 18 x) - \frac{2}{3} \cdot 81 - 2$

Step 2.5.2

Cancel the common factor of $3$ .

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Step 2.5.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - \frac{x^{2} \cdot 2}{3} + \frac{- 2}{3} \cdot (- 18 x) - \frac{2}{3} \cdot 81 - 2$

Step 2.5.2.2

Factor $3$ out of $- 18 x$ .

$y = - \frac{x^{2} \cdot 2}{3} + \frac{- 2}{3} \cdot (3 (- 6 x)) - \frac{2}{3} \cdot 81 - 2$

Step 2.5.2.3

Cancel the common factor.

$y = - \frac{x^{2} \cdot 2}{3} + \frac{- 2}{3} \cdot (3 (- 6 x)) - \frac{2}{3} \cdot 81 - 2$

Step 2.5.2.4

Rewrite the expression.

$y = - \frac{x^{2} \cdot 2}{3} - 2 (- 6 x) - \frac{2}{3} \cdot 81 - 2$

Step 2.5.3

Multiply $- 6$ by $- 2$ .

$y = - \frac{x^{2} \cdot 2}{3} + 12 x - \frac{2}{3} \cdot 81 - 2$

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - \frac{x^{2} \cdot 2}{3} + 12 x + \frac{- 2}{3} \cdot 81 - 2$

Step 2.5.4.2

Factor $3$ out of $81$ .

$y = - \frac{x^{2} \cdot 2}{3} + 12 x + \frac{- 2}{3} \cdot (3 (27)) - 2$

Step 2.5.4.3

Cancel the common factor.

$y = - \frac{x^{2} \cdot 2}{3} + 12 x + \frac{- 2}{3} \cdot (3 \cdot 27) - 2$

Step 2.5.4.4

Rewrite the expression.

$y = - \frac{x^{2} \cdot 2}{3} + 12 x - 2 \cdot 27 - 2$

Step 2.5.5

Multiply $- 2$ by $27$ .

$y = - \frac{x^{2} \cdot 2}{3} + 12 x - 54 - 2$

Step 2.6

Move $2$ to the left of $x^{2}$ .

$y = - \frac{2 x^{2}}{3} + 12 x - 54 - 2$

Step 3

Subtract $2$ from $- 54$ .

$y = - \frac{2 x^{2}}{3} + 12 x - 56$

Step 4

Reorder terms.

$y = - (\frac{2}{3} x^{2}) + 12 x - 56$

Step 5

Remove parentheses.

$y = - \frac{2}{3} x^{2} + 12 x - 56$

Step 6