Pre-Algebra Examples

$7 x (6 x + 8) = - 5$

Step 1

Divide each term in $7 x (6 x + 8) = - 5$ by $7$ and simplify.

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Step 1.1

Divide each term in $7 x (6 x + 8) = - 5$ by $7$ .

$\frac{7 x (6 x + 8)}{7} = \frac{- 5}{7}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify terms.

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Step 1.2.1.1

Cancel the common factor of $7$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{7 x (6 x + 8)}{7} = \frac{- 5}{7}$

Step 1.2.1.1.2

Divide $x (6 x + 8)$ by $1$ .

$x (6 x + 8) = \frac{- 5}{7}$

Step 1.2.1.2

Apply the distributive property.

$x (6 x) + x \cdot 8 = \frac{- 5}{7}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Rewrite using the commutative property of multiplication.

$6 x \cdot x + x \cdot 8 = \frac{- 5}{7}$

Step 1.2.1.3.2

Move $8$ to the left of $x$ .

$6 x \cdot x + 8 \cdot x = \frac{- 5}{7}$

Step 1.2.2

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

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Step 1.2.2.1

Move $x$ .

$6 (x \cdot x) + 8 \cdot x = \frac{- 5}{7}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$6 x^{2} + 8 \cdot x = \frac{- 5}{7}$

$6 x^{2} + 8 x = \frac{- 5}{7}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$6 x^{2} + 8 x = - \frac{5}{7}$

Step 2

Add $\frac{5}{7}$ to both sides of the equation.

$6 x^{2} + 8 x + \frac{5}{7} = 0$

Step 3

Multiply through by the least common denominator $7$ , then simplify.

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Step 3.1

Apply the distributive property.

$7 (6 x^{2}) + 7 (8 x) + 7 (\frac{5}{7}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $6$ by $7$ .

$42 x^{2} + 7 (8 x) + 7 (\frac{5}{7}) = 0$

Step 3.2.2

Multiply $8$ by $7$ .

$42 x^{2} + 56 x + 7 (\frac{5}{7}) = 0$

Step 3.2.3

Cancel the common factor of $7$ .

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Step 3.2.3.1

Cancel the common factor.

$42 x^{2} + 56 x + 7 (\frac{5}{7}) = 0$

Step 3.2.3.2

Rewrite the expression.

$42 x^{2} + 56 x + 5 = 0$

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 $a = 42$ , $b = 56$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 56 \pm \sqrt{56^{2} - 4 \cdot (42 \cdot 5)}}{2 \cdot 42}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $56$ to the power of $2$ .

$x = \frac{- 56 \pm \sqrt{3136 - 4 \cdot 42 \cdot 5}}{2 \cdot 42}$

Step 6.1.2

Multiply $- 4 \cdot 42 \cdot 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $42$ .

$x = \frac{- 56 \pm \sqrt{3136 - 168 \cdot 5}}{2 \cdot 42}$

Step 6.1.2.2

Multiply $- 168$ by $5$ .

$x = \frac{- 56 \pm \sqrt{3136 - 840}}{2 \cdot 42}$

Step 6.1.3

Subtract $840$ from $3136$ .

$x = \frac{- 56 \pm \sqrt{2296}}{2 \cdot 42}$

Step 6.1.4

Rewrite $2296$ as $2^{2} \cdot 574$ .

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Step 6.1.4.1

Factor $4$ out of $2296$ .

$x = \frac{- 56 \pm \sqrt{4 (574)}}{2 \cdot 42}$

Step 6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 56 \pm \sqrt{2^{2} \cdot 574}}{2 \cdot 42}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 56 \pm 2 \sqrt{574}}{2 \cdot 42}$

Step 6.2

Multiply $2$ by $42$ .

$x = \frac{- 56 \pm 2 \sqrt{574}}{84}$

Step 6.3

Simplify $\frac{- 56 \pm 2 \sqrt{574}}{84}$ .

$x = \frac{- 28 \pm \sqrt{574}}{42}$

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 $56$ to the power of $2$ .

$x = \frac{- 56 \pm \sqrt{3136 - 4 \cdot 42 \cdot 5}}{2 \cdot 42}$

Step 7.1.2

Multiply $- 4 \cdot 42 \cdot 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $42$ .

$x = \frac{- 56 \pm \sqrt{3136 - 168 \cdot 5}}{2 \cdot 42}$

Step 7.1.2.2

Multiply $- 168$ by $5$ .

$x = \frac{- 56 \pm \sqrt{3136 - 840}}{2 \cdot 42}$

Step 7.1.3

Subtract $840$ from $3136$ .

$x = \frac{- 56 \pm \sqrt{2296}}{2 \cdot 42}$

Step 7.1.4

Rewrite $2296$ as $2^{2} \cdot 574$ .

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Step 7.1.4.1

Factor $4$ out of $2296$ .

$x = \frac{- 56 \pm \sqrt{4 (574)}}{2 \cdot 42}$

Step 7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 56 \pm \sqrt{2^{2} \cdot 574}}{2 \cdot 42}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 56 \pm 2 \sqrt{574}}{2 \cdot 42}$

Step 7.2

Multiply $2$ by $42$ .

$x = \frac{- 56 \pm 2 \sqrt{574}}{84}$

Step 7.3

Simplify $\frac{- 56 \pm 2 \sqrt{574}}{84}$ .

$x = \frac{- 28 \pm \sqrt{574}}{42}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{- 28 + \sqrt{574}}{42}$

Step 7.5

Rewrite $- 28$ as $- 1 (28)$ .

$x = \frac{- 1 \cdot 28 + \sqrt{574}}{42}$

Step 7.6

Factor $- 1$ out of $\sqrt{574}$ .

$x = \frac{- 1 \cdot 28 - 1 (- \sqrt{574})}{42}$

Step 7.7

Factor $- 1$ out of $- 1 (28) - 1 (- \sqrt{574})$ .

$x = \frac{- 1 (28 - \sqrt{574})}{42}$

Step 7.8

Move the negative in front of the fraction.

$x = - \frac{28 - \sqrt{574}}{42}$

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 $56$ to the power of $2$ .

$x = \frac{- 56 \pm \sqrt{3136 - 4 \cdot 42 \cdot 5}}{2 \cdot 42}$

Step 8.1.2

Multiply $- 4 \cdot 42 \cdot 5$ .

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Step 8.1.2.1

Multiply $- 4$ by $42$ .

$x = \frac{- 56 \pm \sqrt{3136 - 168 \cdot 5}}{2 \cdot 42}$

Step 8.1.2.2

Multiply $- 168$ by $5$ .

$x = \frac{- 56 \pm \sqrt{3136 - 840}}{2 \cdot 42}$

Step 8.1.3

Subtract $840$ from $3136$ .

$x = \frac{- 56 \pm \sqrt{2296}}{2 \cdot 42}$

Step 8.1.4

Rewrite $2296$ as $2^{2} \cdot 574$ .

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Step 8.1.4.1

Factor $4$ out of $2296$ .

$x = \frac{- 56 \pm \sqrt{4 (574)}}{2 \cdot 42}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 56 \pm \sqrt{2^{2} \cdot 574}}{2 \cdot 42}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{- 56 \pm 2 \sqrt{574}}{2 \cdot 42}$

Step 8.2

Multiply $2$ by $42$ .

$x = \frac{- 56 \pm 2 \sqrt{574}}{84}$

Step 8.3

Simplify $\frac{- 56 \pm 2 \sqrt{574}}{84}$ .

$x = \frac{- 28 \pm \sqrt{574}}{42}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{- 28 - \sqrt{574}}{42}$

Step 8.5

Rewrite $- 28$ as $- 1 (28)$ .

$x = \frac{- 1 \cdot 28 - \sqrt{574}}{42}$

Step 8.6

Factor $- 1$ out of $- \sqrt{574}$ .

$x = \frac{- 1 \cdot 28 - (\sqrt{574})}{42}$

Step 8.7

Factor $- 1$ out of $- 1 (28) - (\sqrt{574})$ .

$x = \frac{- 1 (28 + \sqrt{574})}{42}$

Step 8.8

Move the negative in front of the fraction.

$x = - \frac{28 + \sqrt{574}}{42}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{28 - \sqrt{574}}{42}, - \frac{28 + \sqrt{574}}{42}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{28 - \sqrt{574}}{42}, - \frac{28 + \sqrt{574}}{42}$

Decimal Form:

$x = - 0.09623102 \dots, - 1.23710231 \dots$