Basic Math Examples

$(205 - \frac{3}{2} w) w = 5500$

Step 1

Simplify $(205 - \frac{3}{2} w) w$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$(205 - \frac{w \cdot 3}{2}) w = 5500$

Step 1.1.2

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

$(205 - \frac{3 w}{2}) w = 5500$

Step 1.2

Apply the distributive property.

$205 w - \frac{3 w}{2} w = 5500$

Step 1.3

Multiply $- \frac{3 w}{2} w$ .

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Step 1.3.1

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

$205 w - \frac{w (3 w)}{2} = 5500$

Step 1.3.2

Raise $w$ to the power of $1$ .

$205 w - \frac{3 (w^{1} w)}{2} = 5500$

Step 1.3.3

Raise $w$ to the power of $1$ .

$205 w - \frac{3 (w^{1} w^{1})}{2} = 5500$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$205 w - \frac{3 w^{1 + 1}}{2} = 5500$

Step 1.3.5

Add $1$ and $1$ .

$205 w - \frac{3 w^{2}}{2} = 5500$

Step 2

Subtract $5500$ from both sides of the equation.

$205 w - \frac{3 w^{2}}{2} - 5500 = 0$

Step 3

Multiply through by the least common denominator $2$ , then simplify.

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Step 3.1

Apply the distributive property.

$2 (205 w) + 2 (- \frac{3 w^{2}}{2}) + 2 \cdot - 5500 = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $205$ by $2$ .

$410 w + 2 (- \frac{3 w^{2}}{2}) + 2 \cdot - 5500 = 0$

Step 3.2.2

Cancel the common factor of $2$ .

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Step 3.2.2.1

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

$410 w + 2 (\frac{- 3 w^{2}}{2}) + 2 \cdot - 5500 = 0$

Step 3.2.2.2

Cancel the common factor.

$410 w + 2 (\frac{- 3 w^{2}}{2}) + 2 \cdot - 5500 = 0$

Step 3.2.2.3

Rewrite the expression.

$410 w - 3 w^{2} + 2 \cdot - 5500 = 0$

Step 3.2.3

Multiply $2$ by $- 5500$ .

$410 w - 3 w^{2} - 11000 = 0$

Step 3.3

Reorder $410 w$ and $- 3 w^{2}$ .

$- 3 w^{2} + 410 w - 11000 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 3$ , $b = 410$ , and $c = - 11000$ into the quadratic formula and solve for $w$ .

$\frac{- 410 \pm \sqrt{410^{2} - 4 \cdot (- 3 \cdot - 11000)}}{2 \cdot - 3}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $410$ to the power of $2$ .

$w = \frac{- 410 \pm \sqrt{168100 - 4 \cdot - 3 \cdot - 11000}}{2 \cdot - 3}$

Step 6.1.2

Multiply $- 4 \cdot - 3 \cdot - 11000$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 3$ .

$w = \frac{- 410 \pm \sqrt{168100 + 12 \cdot - 11000}}{2 \cdot - 3}$

Step 6.1.2.2

Multiply $12$ by $- 11000$ .

$w = \frac{- 410 \pm \sqrt{168100 - 132000}}{2 \cdot - 3}$

Step 6.1.3

Subtract $132000$ from $168100$ .

$w = \frac{- 410 \pm \sqrt{36100}}{2 \cdot - 3}$

Step 6.1.4

Rewrite $36100$ as $190^{2}$ .

$w = \frac{- 410 \pm \sqrt{190^{2}}}{2 \cdot - 3}$

Step 6.1.5

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

$w = \frac{- 410 \pm 190}{2 \cdot - 3}$

Step 6.2

Multiply $2$ by $- 3$ .

$w = \frac{- 410 \pm 190}{- 6}$

Step 6.3

Simplify $\frac{- 410 \pm 190}{- 6}$ .

$w = \frac{205 \pm 95}{3}$

Step 7

The final answer is the combination of both solutions.

$w = 100, \frac{110}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$w = 100, \frac{110}{3}$

Decimal Form:

$w = 100, 36.\overline{6}$

Mixed Number Form:

$w = 100, 36 \frac{2}{3}$