Precalculus Examples

$w (27 - w) = 120$

Step 1

Simplify $w (27 - w)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$w \cdot 27 + w (- w) = 120$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Move $27$ to the left of $w$ .

$27 \cdot w + w (- w) = 120$

Step 1.1.2.2

Rewrite using the commutative property of multiplication.

$27 \cdot w - w \cdot w = 120$

Step 1.2

Multiply $w$ by $w$ by adding the exponents.

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Step 1.2.1

Move $w$ .

$27 w - (w \cdot w) = 120$

Step 1.2.2

Multiply $w$ by $w$ .

$27 w - w^{2} = 120$

Step 2

Subtract $120$ from both sides of the equation.

$27 w - w^{2} - 120 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 1$ , $b = 27$ , and $c = - 120$ into the quadratic formula and solve for $w$ .

$\frac{- 27 \pm \sqrt{27^{2} - 4 \cdot (- 1 \cdot - 120)}}{2 \cdot - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $27$ to the power of $2$ .

$w = \frac{- 27 \pm \sqrt{729 - 4 \cdot - 1 \cdot - 120}}{2 \cdot - 1}$

Step 5.1.2

Multiply $- 4 \cdot - 1 \cdot - 120$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

$w = \frac{- 27 \pm \sqrt{729 + 4 \cdot - 120}}{2 \cdot - 1}$

Step 5.1.2.2

Multiply $4$ by $- 120$ .

$w = \frac{- 27 \pm \sqrt{729 - 480}}{2 \cdot - 1}$

Step 5.1.3

Subtract $480$ from $729$ .

$w = \frac{- 27 \pm \sqrt{249}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$w = \frac{- 27 \pm \sqrt{249}}{- 2}$

Step 5.3

Simplify $\frac{- 27 \pm \sqrt{249}}{- 2}$ .

$w = \frac{27 \pm \sqrt{249}}{2}$

Step 6

The final answer is the combination of both solutions.

$w = \frac{27 + \sqrt{249}}{2}, \frac{27 - \sqrt{249}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$w = \frac{27 + \sqrt{249}}{2}, \frac{27 - \sqrt{249}}{2}$

Decimal Form:

$w = 21.38986691 \dots, 5.61013308 \dots$