Precalculus Examples

$15 = 0.5 \cdot (x - 1) \cdot x$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $0.5 \cdot (x - 1) \cdot x$ .

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Step 1.1.1.1

Apply the distributive property.

$15 = (0.5 x + 0.5 \cdot - 1) \cdot x$

Step 1.1.1.2

Multiply $0.5$ by $- 1$ .

$15 = (0.5 x - 0.5) \cdot x$

Step 1.1.1.3

Apply the distributive property.

$15 = 0.5 x \cdot x - 0.5 x$

Step 1.1.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.4.1

Move $x$ .

$15 = 0.5 (x \cdot x) - 0.5 x$

Step 1.1.1.4.2

Multiply $x$ by $x$ .

$15 = 0.5 x^{2} - 0.5 x$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Subtract $0.5 x^{2}$ from both sides of the equation.

$15 - 0.5 x^{2} = - 0.5 x$

Step 1.2.2

Add $0.5 x$ to both sides of the equation.

$15 - 0.5 x^{2} + 0.5 x = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = - 0.5$ , $b = 0.5$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{- 0.5 \pm \sqrt{{0.5}^{2} - 4 \cdot (- 0.5 \cdot 15)}}{2 \cdot - 0.5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $0.5$ to the power of $2$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 - 4 \cdot - 0.5 \cdot 15}}{2 \cdot - 0.5}$

Step 4.1.2

Multiply $- 4 \cdot - 0.5 \cdot 15$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 0.5$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 + 2 \cdot 15}}{2 \cdot - 0.5}$

Step 4.1.2.2

Multiply $2$ by $15$ .

$x = \frac{- 0.5 \pm \sqrt{0.25 + 30}}{2 \cdot - 0.5}$

Step 4.1.3

Add $0.25$ and $30$ .

$x = \frac{- 0.5 \pm \sqrt{30.25}}{2 \cdot - 0.5}$

Step 4.1.4

Rewrite $30.25$ as ${5.5}^{2}$ .

$x = \frac{- 0.5 \pm \sqrt{{5.5}^{2}}}{2 \cdot - 0.5}$

Step 4.1.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 0.5 \pm 5.5}{2 \cdot - 0.5}$

Step 4.2

Multiply $2$ by $- 0.5$ .

$x = \frac{- 0.5 \pm 5.5}{- 1}$

Step 4.3

Move the negative one from the denominator of $\frac{- 0.5 \pm 5.5}{- 1}$ .

$x = - 1 \cdot (- 0.5 \pm 5.5)$

Step 4.4

Rewrite $- 1 \cdot (- 0.5 \pm 5.5)$ as $- (- 0.5 \pm 5.5)$ .

$x = - (- 0.5 \pm 5.5)$

Step 5

The final answer is the combination of both solutions.

$x = - 5, 6$