Pre-Algebra Examples

$3 x + 5 x \cdot (408 - x) = 1624$

Step 1

Simplify $3 x + 5 x \cdot (408 - x)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$3 x + 5 x \cdot 408 + 5 x (- x) = 1624$

Step 1.1.2

Multiply $408$ by $5$ .

$3 x + 2040 x + 5 x (- x) = 1624$

Step 1.1.3

Rewrite using the commutative property of multiplication.

$3 x + 2040 x + 5 \cdot - 1 x \cdot x = 1624$

Step 1.1.4

Simplify each term.

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Step 1.1.4.1

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

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Step 1.1.4.1.1

Move $x$ .

$3 x + 2040 x + 5 \cdot - 1 (x \cdot x) = 1624$

Step 1.1.4.1.2

Multiply $x$ by $x$ .

$3 x + 2040 x + 5 \cdot - 1 x^{2} = 1624$

Step 1.1.4.2

Multiply $5$ by $- 1$ .

$3 x + 2040 x - 5 x^{2} = 1624$

Step 1.2

Add $3 x$ and $2040 x$ .

$2043 x - 5 x^{2} = 1624$

Step 2

Subtract $1624$ from both sides of the equation.

$2043 x - 5 x^{2} - 1624 = 0$

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 $a = - 5$ , $b = 2043$ , and $c = - 1624$ into the quadratic formula and solve for $x$ .

$\frac{- 2043 \pm \sqrt{2043^{2} - 4 \cdot (- 5 \cdot - 1624)}}{2 \cdot - 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $2043$ to the power of $2$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 4 \cdot - 5 \cdot - 1624}}{2 \cdot - 5}$

Step 5.1.2

Multiply $- 4 \cdot - 5 \cdot - 1624$ .

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Step 5.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4173849 + 20 \cdot - 1624}}{2 \cdot - 5}$

Step 5.1.2.2

Multiply $20$ by $- 1624$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 32480}}{2 \cdot - 5}$

Step 5.1.3

Subtract $32480$ from $4173849$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{2 \cdot - 5}$

Step 5.2

Multiply $2$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{- 10}$

Step 5.3

Simplify $\frac{- 2043 \pm \sqrt{4141369}}{- 10}$ .

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 5.4

Multiply by $1$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10}$

Step 5.5

Factor $10$ out of $10$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10 (1)}$

Step 5.6

Separate fractions.

$x = \frac{1}{10} \cdot \frac{2043 \pm \sqrt{4141369}}{1}$

Step 5.7

Divide $1$ by $10$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{1})$

Step 5.8

Cancel the common factor of $0.1$ .

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Step 5.8.1

Factor $0.1$ out of $1$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 5.8.2

Cancel the common factor.

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 5.8.3

Rewrite the expression.

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $2043$ to the power of $2$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 4 \cdot - 5 \cdot - 1624}}{2 \cdot - 5}$

Step 6.1.2

Multiply $- 4 \cdot - 5 \cdot - 1624$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4173849 + 20 \cdot - 1624}}{2 \cdot - 5}$

Step 6.1.2.2

Multiply $20$ by $- 1624$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 32480}}{2 \cdot - 5}$

Step 6.1.3

Subtract $32480$ from $4173849$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{2 \cdot - 5}$

Step 6.2

Multiply $2$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{- 10}$

Step 6.3

Simplify $\frac{- 2043 \pm \sqrt{4141369}}{- 10}$ .

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 6.4

Multiply by $1$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10}$

Step 6.5

Factor $10$ out of $10$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10 (1)}$

Step 6.6

Separate fractions.

$x = \frac{1}{10} \cdot \frac{2043 \pm \sqrt{4141369}}{1}$

Step 6.7

Divide $1$ by $10$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{1})$

Step 6.8

Cancel the common factor of $0.1$ .

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Step 6.8.1

Factor $0.1$ out of $1$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 6.8.2

Cancel the common factor.

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 6.8.3

Rewrite the expression.

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 6.9

Change the $\pm$ to $+$ .

$x = \frac{2043 + \sqrt{4141369}}{10}$

Step 6.10

Add $2043$ and $\sqrt{4141369}$ .

$x = \frac{4078.03538052}{10}$

Step 6.11

Divide $4078.03538052$ by $10$ .

$x = 407.80353805$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $2043$ to the power of $2$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 4 \cdot - 5 \cdot - 1624}}{2 \cdot - 5}$

Step 7.1.2

Multiply $- 4 \cdot - 5 \cdot - 1624$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4173849 + 20 \cdot - 1624}}{2 \cdot - 5}$

Step 7.1.2.2

Multiply $20$ by $- 1624$ .

$x = \frac{- 2043 \pm \sqrt{4173849 - 32480}}{2 \cdot - 5}$

Step 7.1.3

Subtract $32480$ from $4173849$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{2 \cdot - 5}$

Step 7.2

Multiply $2$ by $- 5$ .

$x = \frac{- 2043 \pm \sqrt{4141369}}{- 10}$

Step 7.3

Simplify $\frac{- 2043 \pm \sqrt{4141369}}{- 10}$ .

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 7.4

Multiply by $1$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10}$

Step 7.5

Factor $10$ out of $10$ .

$x = \frac{1 (2043 \pm \sqrt{4141369})}{10 (1)}$

Step 7.6

Separate fractions.

$x = \frac{1}{10} \cdot \frac{2043 \pm \sqrt{4141369}}{1}$

Step 7.7

Divide $1$ by $10$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{1})$

Step 7.8

Cancel the common factor of $0.1$ .

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Step 7.8.1

Factor $0.1$ out of $1$ .

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 7.8.2

Cancel the common factor.

$x = 0.1 (\frac{2043 \pm \sqrt{4141369}}{0.1 \cdot 10})$

Step 7.8.3

Rewrite the expression.

$x = \frac{2043 \pm \sqrt{4141369}}{10}$

Step 7.9

Change the $\pm$ to $-$ .

$x = \frac{2043 - \sqrt{4141369}}{10}$

Step 7.10

Subtract $\sqrt{4141369}$ from $2043$ .

$x = \frac{7.96461947}{10}$

Step 7.11

Divide $7.96461947$ by $10$ .

$x = 0.79646194$

Step 8

The final answer is the combination of both solutions.

$x = 407.80353805, 0.79646194$