Pre-Algebra Examples

$0.5 = 500 x^{2} + 20 x + 600$

Step 1

Rewrite the equation as $500 x^{2} + 20 x + 600 = 0.5$ .

$500 x^{2} + 20 x + 600 = 0.5$

Step 2

Subtract $0.5$ from both sides of the equation.

$500 x^{2} + 20 x + 600 - 0.5 = 0$

Step 3

Subtract $0.5$ from $600$ .

$500 x^{2} + 20 x + 599.5 = 0$

Step 4

Factor $0.5$ out of $500 x^{2} + 20 x + 599.5$ .

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Step 4.1

Factor $0.5$ out of $500 x^{2}$ .

$0.5 (1000 x^{2}) + 20 x + 599.5 = 0$

Step 4.2

Factor $0.5$ out of $20 x$ .

$0.5 (1000 x^{2}) + 0.5 (40 x) + 599.5 = 0$

Step 4.3

Factor $0.5$ out of $599.5$ .

$0.5 (1000 x^{2}) + 0.5 (40 x) + 0.5 (1199) = 0$

Step 4.4

Factor $0.5$ out of $0.5 (1000 x^{2}) + 0.5 (40 x)$ .

$0.5 (1000 x^{2} + 40 x) + 0.5 (1199) = 0$

Step 4.5

Factor $0.5$ out of $0.5 (1000 x^{2} + 40 x) + 0.5 (1199)$ .

$0.5 (1000 x^{2} + 40 x + 1199) = 0$

Step 5

Divide each term in $0.5 (1000 x^{2} + 40 x + 1199) = 0$ by $0.5$ and simplify.

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Step 5.1

Divide each term in $0.5 (1000 x^{2} + 40 x + 1199) = 0$ by $0.5$ .

$\frac{0.5 (1000 x^{2} + 40 x + 1199)}{0.5} = \frac{0}{0.5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $0.5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{0.5 (1000 x^{2} + 40 x + 1199)}{0.5} = \frac{0}{0.5}$

Step 5.2.1.2

Divide $1000 x^{2} + 40 x + 1199$ by $1$ .

$1000 x^{2} + 40 x + 1199 = \frac{0}{0.5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $0.5$ .

$1000 x^{2} + 40 x + 1199 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1000$ , $b = 40$ , and $c = 1199$ into the quadratic formula and solve for $x$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (1000 \cdot 1199)}}{2 \cdot 1000}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 1000 \cdot 1199}}{2 \cdot 1000}$

Step 8.1.2

Multiply $- 4 \cdot 1000 \cdot 1199$ .

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Step 8.1.2.1

Multiply $- 4$ by $1000$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4000 \cdot 1199}}{2 \cdot 1000}$

Step 8.1.2.2

Multiply $- 4000$ by $1199$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4796000}}{2 \cdot 1000}$

Step 8.1.3

Subtract $4796000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{- 4794400}}{2 \cdot 1000}$

Step 8.1.4

Rewrite $- 4794400$ as $- 1 (4794400)$ .

$x = \frac{- 40 \pm \sqrt{- 1 \cdot 4794400}}{2 \cdot 1000}$

Step 8.1.5

Rewrite $\sqrt{- 1 (4794400)}$ as $\sqrt{- 1} \cdot \sqrt{4794400}$ .

$x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 8.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 40 \pm i \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 8.1.7

Rewrite $4794400$ as $20^{2} \cdot 11986$ .

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Step 8.1.7.1

Factor $400$ out of $4794400$ .

$x = \frac{- 40 \pm i \cdot \sqrt{400 (11986)}}{2 \cdot 1000}$

Step 8.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 40 \pm i \cdot \sqrt{20^{2} \cdot 11986}}{2 \cdot 1000}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{- 40 \pm i \cdot (20 \sqrt{11986})}{2 \cdot 1000}$

Step 8.1.9

Move $20$ to the left of $i$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2 \cdot 1000}$

Step 8.2

Multiply $2$ by $1000$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2000}$

Step 8.3

Simplify $\frac{- 40 \pm 20 i \sqrt{11986}}{2000}$ .

$x = \frac{- 2 \pm i \sqrt{11986}}{100}$

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 $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 1000 \cdot 1199}}{2 \cdot 1000}$

Step 9.1.2

Multiply $- 4 \cdot 1000 \cdot 1199$ .

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Step 9.1.2.1

Multiply $- 4$ by $1000$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4000 \cdot 1199}}{2 \cdot 1000}$

Step 9.1.2.2

Multiply $- 4000$ by $1199$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4796000}}{2 \cdot 1000}$

Step 9.1.3

Subtract $4796000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{- 4794400}}{2 \cdot 1000}$

Step 9.1.4

Rewrite $- 4794400$ as $- 1 (4794400)$ .

$x = \frac{- 40 \pm \sqrt{- 1 \cdot 4794400}}{2 \cdot 1000}$

Step 9.1.5

Rewrite $\sqrt{- 1 (4794400)}$ as $\sqrt{- 1} \cdot \sqrt{4794400}$ .

$x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 9.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 40 \pm i \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 9.1.7

Rewrite $4794400$ as $20^{2} \cdot 11986$ .

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Step 9.1.7.1

Factor $400$ out of $4794400$ .

$x = \frac{- 40 \pm i \cdot \sqrt{400 (11986)}}{2 \cdot 1000}$

Step 9.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 40 \pm i \cdot \sqrt{20^{2} \cdot 11986}}{2 \cdot 1000}$

Step 9.1.8

Pull terms out from under the radical.

$x = \frac{- 40 \pm i \cdot (20 \sqrt{11986})}{2 \cdot 1000}$

Step 9.1.9

Move $20$ to the left of $i$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2 \cdot 1000}$

Step 9.2

Multiply $2$ by $1000$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2000}$

Step 9.3

Simplify $\frac{- 40 \pm 20 i \sqrt{11986}}{2000}$ .

$x = \frac{- 2 \pm i \sqrt{11986}}{100}$

Step 9.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + i \sqrt{11986}}{100}$

Step 9.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + i \sqrt{11986}}{100}$

Step 9.6

Factor $- 1$ out of $i \sqrt{11986}$ .

$x = \frac{- 1 \cdot 2 - (- i \sqrt{11986})}{100}$

Step 9.7

Factor $- 1$ out of $- 1 (2) - (- i \sqrt{11986})$ .

$x = \frac{- 1 (2 - i \sqrt{11986})}{100}$

Step 9.8

Move the negative in front of the fraction.

$x = - \frac{2 - i \sqrt{11986}}{100}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 1000 \cdot 1199}}{2 \cdot 1000}$

Step 10.1.2

Multiply $- 4 \cdot 1000 \cdot 1199$ .

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Step 10.1.2.1

Multiply $- 4$ by $1000$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4000 \cdot 1199}}{2 \cdot 1000}$

Step 10.1.2.2

Multiply $- 4000$ by $1199$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4796000}}{2 \cdot 1000}$

Step 10.1.3

Subtract $4796000$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{- 4794400}}{2 \cdot 1000}$

Step 10.1.4

Rewrite $- 4794400$ as $- 1 (4794400)$ .

$x = \frac{- 40 \pm \sqrt{- 1 \cdot 4794400}}{2 \cdot 1000}$

Step 10.1.5

Rewrite $\sqrt{- 1 (4794400)}$ as $\sqrt{- 1} \cdot \sqrt{4794400}$ .

$x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 10.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 40 \pm i \cdot \sqrt{4794400}}{2 \cdot 1000}$

Step 10.1.7

Rewrite $4794400$ as $20^{2} \cdot 11986$ .

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Step 10.1.7.1

Factor $400$ out of $4794400$ .

$x = \frac{- 40 \pm i \cdot \sqrt{400 (11986)}}{2 \cdot 1000}$

Step 10.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{- 40 \pm i \cdot \sqrt{20^{2} \cdot 11986}}{2 \cdot 1000}$

Step 10.1.8

Pull terms out from under the radical.

$x = \frac{- 40 \pm i \cdot (20 \sqrt{11986})}{2 \cdot 1000}$

Step 10.1.9

Move $20$ to the left of $i$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2 \cdot 1000}$

Step 10.2

Multiply $2$ by $1000$ .

$x = \frac{- 40 \pm 20 i \sqrt{11986}}{2000}$

Step 10.3

Simplify $\frac{- 40 \pm 20 i \sqrt{11986}}{2000}$ .

$x = \frac{- 2 \pm i \sqrt{11986}}{100}$

Step 10.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - i \sqrt{11986}}{100}$

Step 10.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - i \sqrt{11986}}{100}$

Step 10.6

Factor $- 1$ out of $- i \sqrt{11986}$ .

$x = \frac{- 1 \cdot 2 - (i \sqrt{11986})}{100}$

Step 10.7

Factor $- 1$ out of $- 1 (2) - (i \sqrt{11986})$ .

$x = \frac{- 1 (2 + i \sqrt{11986})}{100}$

Step 10.8

Move the negative in front of the fraction.

$x = - \frac{2 + i \sqrt{11986}}{100}$

Step 11

The final answer is the combination of both solutions.

$x = - \frac{2 - i \sqrt{11986}}{100}, - \frac{2 + i \sqrt{11986}}{100}$