Algebra Examples

$f (x) = - \frac{3}{2} x^{2} + 6 x + \frac{15}{2}$

Step 1

Set $- \frac{3}{2} x^{2} + 6 x + \frac{15}{2}$ equal to $0$ .

$- \frac{3}{2} x^{2} + 6 x + \frac{15}{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$- \frac{x^{2} \cdot 3}{2} + 6 x + \frac{15}{2} = 0$

Step 2.1.2

Move $3$ to the left of $x^{2}$ .

$- \frac{3 x^{2}}{2} + 6 x + \frac{15}{2} = 0$

Step 2.2

Multiply each term in $- \frac{3 x^{2}}{2} + 6 x + \frac{15}{2} = 0$ by $2$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $- \frac{3 x^{2}}{2} + 6 x + \frac{15}{2} = 0$ by $2$ .

$- \frac{3 x^{2}}{2} \cdot 2 + 6 x \cdot 2 + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.1

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

$\frac{- 3 x^{2}}{2} \cdot 2 + 6 x \cdot 2 + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.1.2

Cancel the common factor.

$\frac{- 3 x^{2}}{2} \cdot 2 + 6 x \cdot 2 + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.1.3

Rewrite the expression.

$- 3 x^{2} + 6 x \cdot 2 + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.2

Multiply $2$ by $6$ .

$- 3 x^{2} + 12 x + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.3

Cancel the common factor of $2$ .

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Step 2.2.2.1.3.1

Cancel the common factor.

$- 3 x^{2} + 12 x + \frac{15}{2} \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.3.2

Rewrite the expression.

$- 3 x^{2} + 12 x + 15 = 0 \cdot 2$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $2$ .

$- 3 x^{2} + 12 x + 15 = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 3$ out of $- 3 x^{2} + 12 x + 15$ .

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Step 2.3.1.1

Factor $- 3$ out of $- 3 x^{2}$ .

$- 3 x^{2} + 12 x + 15 = 0$

Step 2.3.1.2

Factor $- 3$ out of $12 x$ .

$- 3 x^{2} - 3 (- 4 x) + 15 = 0$

Step 2.3.1.3

Factor $- 3$ out of $15$ .

$- 3 x^{2} - 3 (- 4 x) - 3 \cdot - 5 = 0$

Step 2.3.1.4

Factor $- 3$ out of $- 3 (x^{2}) - 3 (- 4 x)$ .

$- 3 (x^{2} - 4 x) - 3 \cdot - 5 = 0$

Step 2.3.1.5

Factor $- 3$ out of $- 3 (x^{2} - 4 x) - 3 (- 5)$ .

$- 3 (x^{2} - 4 x - 5) = 0$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 2.3.2.1.2

Write the factored form using these integers.

$- 3 ((x - 5) (x + 1)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$- 3 (x - 5) (x + 1) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x + 1 = 0$

Step 2.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.7

The final solution is all the values that make $- 3 (x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 3