Algebra Examples

$- 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 1

Write $- 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$ as an equation.

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

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$0 = - 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 2.2.2

Simplify $- 2 {(x - \frac{3}{2})}^{2} + \frac{27}{2}$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Rewrite ${(x - \frac{3}{2})}^{2}$ as $(x - \frac{3}{2}) (x - \frac{3}{2})$ .

$0 = - 2 ((x - \frac{3}{2}) (x - \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.2

Expand $(x - \frac{3}{2}) (x - \frac{3}{2})$ using the FOIL Method.

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Step 2.2.2.1.2.1

Apply the distributive property.

$0 = - 2 (x (x - \frac{3}{2}) - \frac{3}{2} (x - \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.2.2

Apply the distributive property.

$0 = - 2 (x \cdot x + x (- \frac{3}{2}) - \frac{3}{2} (x - \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.2.3

Apply the distributive property.

$0 = - 2 (x \cdot x + x (- \frac{3}{2}) - \frac{3}{2} x - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3

Simplify and combine like terms.

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Step 2.2.2.1.3.1

Simplify each term.

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Step 2.2.2.1.3.1.1

Multiply $x$ by $x$ .

$0 = - 2 (x^{2} + x (- \frac{3}{2}) - \frac{3}{2} x - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3.1.2

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

$0 = - 2 (x^{2} - \frac{x \cdot 3}{2} - \frac{3}{2} x - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3.1.3

Move $3$ to the left of $x$ .

$0 = - 2 (x^{2} - \frac{3 \cdot x}{2} - \frac{3}{2} x - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3.1.4

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

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{x \cdot 3}{2} - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3.1.5

Move $3$ to the left of $x$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 \cdot x}{2} - \frac{3}{2} (- \frac{3}{2})) + \frac{27}{2}$

Step 2.2.2.1.3.1.6

Multiply $- \frac{3}{2} (- \frac{3}{2})$ .

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Step 2.2.2.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + 1 (\frac{3}{2}) \frac{3}{2}) + \frac{27}{2}$

Step 2.2.2.1.3.1.6.2

Multiply $\frac{3}{2}$ by $1$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3}{2} \cdot \frac{3}{2}) + \frac{27}{2}$

Step 2.2.2.1.3.1.6.3

Multiply $\frac{3}{2}$ by $\frac{3}{2}$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3 \cdot 3}{2 \cdot 2}) + \frac{27}{2}$

Step 2.2.2.1.3.1.6.4

Multiply $3$ by $3$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{9}{2 \cdot 2}) + \frac{27}{2}$

Step 2.2.2.1.3.1.6.5

Multiply $2$ by $2$ .

$0 = - 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.3.2

Subtract $\frac{3 x}{2}$ from $- \frac{3 x}{2}$ .

$0 = - 2 (x^{2} - 2 \frac{3 x}{2} + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.4

Simplify each term.

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Step 2.2.2.1.4.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.4.1.1

Factor $2$ out of $- 2$ .

$0 = - 2 (x^{2} + 2 (- 1) \frac{3 x}{2} + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.4.1.2

Cancel the common factor.

$0 = - 2 (x^{2} + 2 \cdot - 1 \frac{3 x}{2} + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.4.1.3

Rewrite the expression.

$0 = - 2 (x^{2} - 1 (3 x) + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.4.2

Multiply $3$ by $- 1$ .

$0 = - 2 (x^{2} - 3 x + \frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.5

Apply the distributive property.

$0 = - 2 x^{2} - 2 (- 3 x) - 2 (\frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.6

Simplify.

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Step 2.2.2.1.6.1

Multiply $- 3$ by $- 2$ .

$0 = - 2 x^{2} + 6 x - 2 (\frac{9}{4}) + \frac{27}{2}$

Step 2.2.2.1.6.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.6.2.1

Factor $2$ out of $- 2$ .

$0 = - 2 x^{2} + 6 x + 2 (- 1) \frac{9}{4} + \frac{27}{2}$

Step 2.2.2.1.6.2.2

Factor $2$ out of $4$ .

$0 = - 2 x^{2} + 6 x + 2 \cdot - 1 \frac{9}{2 \cdot 2} + \frac{27}{2}$

Step 2.2.2.1.6.2.3

Cancel the common factor.

$0 = - 2 x^{2} + 6 x + 2 \cdot - 1 \frac{9}{2 \cdot 2} + \frac{27}{2}$

Step 2.2.2.1.6.2.4

Rewrite the expression.

$0 = - 2 x^{2} + 6 x - 1 (\frac{9}{2}) + \frac{27}{2}$

Step 2.2.2.1.7

Rewrite $- 1 (\frac{9}{2})$ as $- (\frac{9}{2})$ .

$0 = - 2 x^{2} + 6 x - \frac{9}{2} + \frac{27}{2}$

Step 2.2.2.2

Combine fractions.

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Step 2.2.2.2.1

Combine the numerators over the common denominator.

$0 = - 2 x^{2} + 6 x + \frac{- 9 + 27}{2}$

Step 2.2.2.2.2

Simplify the expression.

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Step 2.2.2.2.2.1

Add $- 9$ and $27$ .

$0 = - 2 x^{2} + 6 x + \frac{18}{2}$

Step 2.2.2.2.2.2

Divide $18$ by $2$ .

$0 = - 2 x^{2} + 6 x + 9$

Step 2.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 1.09807621, 4.09807621$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 1.09807621, 0), (4.09807621, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 2 {((0) - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - 2 {(0 - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 3.2.2

Remove parentheses.

$y = - 2 {((0) - \frac{3}{2})}^{2} + \frac{27}{2}$

Step 3.2.3

Simplify $- 2 {((0) - \frac{3}{2})}^{2} + \frac{27}{2}$ .

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Subtract $\frac{3}{2}$ from $0$ .

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

Step 3.2.3.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.3.1.2.1

Apply the product rule to $- \frac{3}{2}$ .

$y = - 2 ({(- 1)}^{2} {(\frac{3}{2})}^{2}) + \frac{27}{2}$

Step 3.2.3.1.2.2

Apply the product rule to $\frac{3}{2}$ .

$y = - 2 ({(- 1)}^{2} \frac{3^{2}}{2^{2}}) + \frac{27}{2}$

Step 3.2.3.1.3

Raise $- 1$ to the power of $2$ .

$y = - 2 (1 \frac{3^{2}}{2^{2}}) + \frac{27}{2}$

Step 3.2.3.1.4

Multiply $\frac{3^{2}}{2^{2}}$ by $1$ .

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

Step 3.2.3.1.5

Raise $3$ to the power of $2$ .

$y = - 2 \frac{9}{2^{2}} + \frac{27}{2}$

Step 3.2.3.1.6

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

$y = - 2 (\frac{9}{4}) + \frac{27}{2}$

Step 3.2.3.1.7

Cancel the common factor of $2$ .

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Step 3.2.3.1.7.1

Factor $2$ out of $- 2$ .

$y = 2 (- 1) \frac{9}{4} + \frac{27}{2}$

Step 3.2.3.1.7.2

Factor $2$ out of $4$ .

$y = 2 \cdot - 1 \frac{9}{2 \cdot 2} + \frac{27}{2}$

Step 3.2.3.1.7.3

Cancel the common factor.

$y = 2 \cdot - 1 \frac{9}{2 \cdot 2} + \frac{27}{2}$

Step 3.2.3.1.7.4

Rewrite the expression.

$y = - 1 (\frac{9}{2}) + \frac{27}{2}$

Step 3.2.3.1.8

Rewrite $- 1 (\frac{9}{2})$ as $- (\frac{9}{2})$ .

$y = - \frac{9}{2} + \frac{27}{2}$

Step 3.2.3.2

Combine fractions.

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Step 3.2.3.2.1

Combine the numerators over the common denominator.

$y = \frac{- 9 + 27}{2}$

Step 3.2.3.2.2

Simplify the expression.

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Step 3.2.3.2.2.1

Add $- 9$ and $27$ .

$y = \frac{18}{2}$

Step 3.2.3.2.2.2

Divide $18$ by $2$ .

$y = 9$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 9)$

Step 4

List the intersections.

x-intercept(s): $(- 1.09807621, 0), (4.09807621, 0)$

y-intercept(s): $(0, 9)$

Step 5