Pre-Algebra Examples

$- 3 x (- (5 x + 3)) - 6 x = 75$

Step 1

Simplify $- 3 x (- (5 x + 3)) - 6 x$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- 3 x (- (5 x) - 1 \cdot 3) - 6 x = 75$

Step 1.1.2

Multiply $5$ by $- 1$ .

$- 3 x (- 5 x - 1 \cdot 3) - 6 x = 75$

Step 1.1.3

Multiply $- 1$ by $3$ .

$- 3 x (- 5 x - 3) - 6 x = 75$

Step 1.1.4

Apply the distributive property.

$- 3 x (- 5 x) - 3 x \cdot - 3 - 6 x = 75$

Step 1.1.5

Rewrite using the commutative property of multiplication.

$- 3 \cdot - 5 x \cdot x - 3 x \cdot - 3 - 6 x = 75$

Step 1.1.6

Multiply $- 3$ by $- 3$ .

$- 3 \cdot - 5 x \cdot x + 9 x - 6 x = 75$

Step 1.1.7

Simplify each term.

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Step 1.1.7.1

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

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Step 1.1.7.1.1

Move $x$ .

$- 3 \cdot - 5 (x \cdot x) + 9 x - 6 x = 75$

Step 1.1.7.1.2

Multiply $x$ by $x$ .

$- 3 \cdot - 5 x^{2} + 9 x - 6 x = 75$

Step 1.1.7.2

Multiply $- 3$ by $- 5$ .

$15 x^{2} + 9 x - 6 x = 75$

Step 1.2

Subtract $6 x$ from $9 x$ .

$15 x^{2} + 3 x = 75$

Step 2

Subtract $75$ from both sides of the equation.

$15 x^{2} + 3 x - 75 = 0$

Step 3

Factor $3$ out of $15 x^{2} + 3 x - 75$ .

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Step 3.1

Factor $3$ out of $15 x^{2}$ .

$3 (5 x^{2}) + 3 x - 75 = 0$

Step 3.2

Factor $3$ out of $3 x$ .

$3 (5 x^{2}) + 3 (x) - 75 = 0$

Step 3.3

Factor $3$ out of $- 75$ .

$3 (5 x^{2}) + 3 x + 3 \cdot - 25 = 0$

Step 3.4

Factor $3$ out of $3 (5 x^{2}) + 3 x$ .

$3 (5 x^{2} + x) + 3 \cdot - 25 = 0$

Step 3.5

Factor $3$ out of $3 (5 x^{2} + x) + 3 \cdot - 25$ .

$3 (5 x^{2} + x - 25) = 0$

Step 4

Divide each term in $3 (5 x^{2} + x - 25) = 0$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 (5 x^{2} + x - 25) = 0$ by $3$ .

$\frac{3 (5 x^{2} + x - 25)}{3} = \frac{0}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 (5 x^{2} + x - 25)}{3} = \frac{0}{3}$

Step 4.2.1.2

Divide $5 x^{2} + x - 25$ by $1$ .

$5 x^{2} + x - 25 = \frac{0}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $3$ .

$5 x^{2} + x - 25 = 0$

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 $a = 5$ , $b = 1$ , and $c = - 25$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (5 \cdot - 25)}}{2 \cdot 5}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 25}}{2 \cdot 5}$

Step 7.1.2

Multiply $- 4 \cdot 5 \cdot - 25$ .

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Step 7.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 25}}{2 \cdot 5}$

Step 7.1.2.2

Multiply $- 20$ by $- 25$ .

$x = \frac{- 1 \pm \sqrt{1 + 500}}{2 \cdot 5}$

Step 7.1.3

Add $1$ and $500$ .

$x = \frac{- 1 \pm \sqrt{501}}{2 \cdot 5}$

Step 7.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{501}}{10}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 25}}{2 \cdot 5}$

Step 8.1.2

Multiply $- 4 \cdot 5 \cdot - 25$ .

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Step 8.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 25}}{2 \cdot 5}$

Step 8.1.2.2

Multiply $- 20$ by $- 25$ .

$x = \frac{- 1 \pm \sqrt{1 + 500}}{2 \cdot 5}$

Step 8.1.3

Add $1$ and $500$ .

$x = \frac{- 1 \pm \sqrt{501}}{2 \cdot 5}$

Step 8.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{501}}{10}$

Step 8.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{501}}{10}$

Step 8.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{501}}{10}$

Step 8.5

Factor $- 1$ out of $\sqrt{501}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{501})}{10}$

Step 8.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{501})$ .

$x = \frac{- 1 (1 - \sqrt{501})}{10}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{501}}{10}$

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

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 25}}{2 \cdot 5}$

Step 9.1.2

Multiply $- 4 \cdot 5 \cdot - 25$ .

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Step 9.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 1 \pm \sqrt{1 - 20 \cdot - 25}}{2 \cdot 5}$

Step 9.1.2.2

Multiply $- 20$ by $- 25$ .

$x = \frac{- 1 \pm \sqrt{1 + 500}}{2 \cdot 5}$

Step 9.1.3

Add $1$ and $500$ .

$x = \frac{- 1 \pm \sqrt{501}}{2 \cdot 5}$

Step 9.2

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm \sqrt{501}}{10}$

Step 9.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{501}}{10}$

Step 9.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{501}}{10}$

Step 9.5

Factor $- 1$ out of $- \sqrt{501}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{501})}{10}$

Step 9.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{501})$ .

$x = \frac{- 1 (1 + \sqrt{501})}{10}$

Step 9.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{501}}{10}$

Step 10

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{501}}{10}, - \frac{1 + \sqrt{501}}{10}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1 - \sqrt{501}}{10}, - \frac{1 + \sqrt{501}}{10}$

Decimal Form:

$x = 2.13830292 \dots, - 2.33830292 \dots$