Pre-Algebra Examples

$20 (20 - x) + \frac{x^{2}}{2} + 168 = 20 \cdot 20$

Step 1

Simplify $20 (20 - x) + \frac{x^{2}}{2} + 168$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$20 \cdot 20 + 20 (- x) + \frac{x^{2}}{2} + 168 = 20 \cdot 20$

Step 1.1.2

Multiply $20$ by $20$ .

$400 + 20 (- x) + \frac{x^{2}}{2} + 168 = 20 \cdot 20$

Step 1.1.3

Multiply $- 1$ by $20$ .

$400 - 20 x + \frac{x^{2}}{2} + 168 = 20 \cdot 20$

Step 1.2

Add $400$ and $168$ .

$- 20 x + \frac{x^{2}}{2} + 568 = 20 \cdot 20$

Step 2

Multiply $20$ by $20$ .

$- 20 x + \frac{x^{2}}{2} + 568 = 400$

Step 3

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

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Step 3.1

Subtract $400$ from both sides of the equation.

$- 20 x + \frac{x^{2}}{2} + 568 - 400 = 0$

Step 3.2

Subtract $400$ from $568$ .

$- 20 x + \frac{x^{2}}{2} + 168 = 0$

Step 4

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

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Step 4.1

Apply the distributive property.

$2 (- 20 x) + 2 (\frac{x^{2}}{2}) + 2 \cdot 168 = 0$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $- 20$ by $2$ .

$- 40 x + 2 (\frac{x^{2}}{2}) + 2 \cdot 168 = 0$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

$- 40 x + 2 (\frac{x^{2}}{2}) + 2 \cdot 168 = 0$

Step 4.2.2.2

Rewrite the expression.

$- 40 x + x^{2} + 2 \cdot 168 = 0$

Step 4.2.3

Multiply $2$ by $168$ .

$- 40 x + x^{2} + 336 = 0$

Step 4.3

Reorder $- 40 x$ and $x^{2}$ .

$x^{2} - 40 x + 336 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 40$ , and $c = 336$ into the quadratic formula and solve for $x$ .

$\frac{40 \pm \sqrt{{(- 40)}^{2} - 4 \cdot (1 \cdot 336)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 40$ to the power of $2$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 1 \cdot 336}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 336$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 336}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $336$ .

$x = \frac{40 \pm \sqrt{1600 - 1344}}{2 \cdot 1}$

Step 7.1.3

Subtract $1344$ from $1600$ .

$x = \frac{40 \pm \sqrt{256}}{2 \cdot 1}$

Step 7.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{40 \pm \sqrt{16^{2}}}{2 \cdot 1}$

Step 7.1.5

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

$x = \frac{40 \pm 16}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{40 \pm 16}{2}$

Step 7.3

Simplify $\frac{40 \pm 16}{2}$ .

$x = 20 \pm 8$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 40$ to the power of $2$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 1 \cdot 336}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 336$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 336}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $336$ .

$x = \frac{40 \pm \sqrt{1600 - 1344}}{2 \cdot 1}$

Step 8.1.3

Subtract $1344$ from $1600$ .

$x = \frac{40 \pm \sqrt{256}}{2 \cdot 1}$

Step 8.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{40 \pm \sqrt{16^{2}}}{2 \cdot 1}$

Step 8.1.5

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

$x = \frac{40 \pm 16}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{40 \pm 16}{2}$

Step 8.3

Simplify $\frac{40 \pm 16}{2}$ .

$x = 20 \pm 8$

Step 8.4

Change the $\pm$ to $+$ .

$x = 20 + 8$

Step 8.5

Add $20$ and $8$ .

$x = 28$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 40$ to the power of $2$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 1 \cdot 336}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot 336$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 336}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $336$ .

$x = \frac{40 \pm \sqrt{1600 - 1344}}{2 \cdot 1}$

Step 9.1.3

Subtract $1344$ from $1600$ .

$x = \frac{40 \pm \sqrt{256}}{2 \cdot 1}$

Step 9.1.4

Rewrite $256$ as $16^{2}$ .

$x = \frac{40 \pm \sqrt{16^{2}}}{2 \cdot 1}$

Step 9.1.5

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

$x = \frac{40 \pm 16}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{40 \pm 16}{2}$

Step 9.3

Simplify $\frac{40 \pm 16}{2}$ .

$x = 20 \pm 8$

Step 9.4

Change the $\pm$ to $-$ .

$x = 20 - 8$

Step 9.5

Subtract $8$ from $20$ .

$x = 12$

Step 10

The final answer is the combination of both solutions.

$x = 28, 12$