Pre-Algebra Examples

$2 x (4 x + 15) = 27$

Step 1

Divide each term in $2 x (4 x + 15) = 27$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 x (4 x + 15) = 27$ by $2$ .

$\frac{2 x (4 x + 15)}{2} = \frac{27}{2}$

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 $2$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{2 x (4 x + 15)}{2} = \frac{27}{2}$

Step 1.2.1.1.2

Divide $x (4 x + 15)$ by $1$ .

$x (4 x + 15) = \frac{27}{2}$

Step 1.2.1.2

Apply the distributive property.

$x (4 x) + x \cdot 15 = \frac{27}{2}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Rewrite using the commutative property of multiplication.

$4 x \cdot x + x \cdot 15 = \frac{27}{2}$

Step 1.2.1.3.2

Move $15$ to the left of $x$ .

$4 x \cdot x + 15 \cdot x = \frac{27}{2}$

Step 1.2.2

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

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Step 1.2.2.1

Move $x$ .

$4 (x \cdot x) + 15 \cdot x = \frac{27}{2}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$4 x^{2} + 15 \cdot x = \frac{27}{2}$

$4 x^{2} + 15 x = \frac{27}{2}$

Step 2

Subtract $\frac{27}{2}$ from both sides of the equation.

$4 x^{2} + 15 x - \frac{27}{2} = 0$

Step 3

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

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Step 3.1

Apply the distributive property.

$2 (4 x^{2}) + 2 (15 x) + 2 (- \frac{27}{2}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $4$ by $2$ .

$8 x^{2} + 2 (15 x) + 2 (- \frac{27}{2}) = 0$

Step 3.2.2

Multiply $15$ by $2$ .

$8 x^{2} + 30 x + 2 (- \frac{27}{2}) = 0$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Move the leading negative in $- \frac{27}{2}$ into the numerator.

$8 x^{2} + 30 x + 2 (\frac{- 27}{2}) = 0$

Step 3.2.3.2

Cancel the common factor.

$8 x^{2} + 30 x + 2 (\frac{- 27}{2}) = 0$

Step 3.2.3.3

Rewrite the expression.

$8 x^{2} + 30 x - 27 = 0$

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 $a = 8$ , $b = 30$ , and $c = - 27$ into the quadratic formula and solve for $x$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (8 \cdot - 27)}}{2 \cdot 8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 8 \cdot - 27}}{2 \cdot 8}$

Step 6.1.2

Multiply $- 4 \cdot 8 \cdot - 27$ .

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Step 6.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{- 30 \pm \sqrt{900 - 32 \cdot - 27}}{2 \cdot 8}$

Step 6.1.2.2

Multiply $- 32$ by $- 27$ .

$x = \frac{- 30 \pm \sqrt{900 + 864}}{2 \cdot 8}$

Step 6.1.3

Add $900$ and $864$ .

$x = \frac{- 30 \pm \sqrt{1764}}{2 \cdot 8}$

Step 6.1.4

Rewrite $1764$ as $42^{2}$ .

$x = \frac{- 30 \pm \sqrt{42^{2}}}{2 \cdot 8}$

Step 6.1.5

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

$x = \frac{- 30 \pm 42}{2 \cdot 8}$

Step 6.2

Multiply $2$ by $8$ .

$x = \frac{- 30 \pm 42}{16}$

Step 6.3

Simplify $\frac{- 30 \pm 42}{16}$ .

$x = \frac{- 15 \pm 21}{8}$

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 $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 8 \cdot - 27}}{2 \cdot 8}$

Step 7.1.2

Multiply $- 4 \cdot 8 \cdot - 27$ .

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Step 7.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{- 30 \pm \sqrt{900 - 32 \cdot - 27}}{2 \cdot 8}$

Step 7.1.2.2

Multiply $- 32$ by $- 27$ .

$x = \frac{- 30 \pm \sqrt{900 + 864}}{2 \cdot 8}$

Step 7.1.3

Add $900$ and $864$ .

$x = \frac{- 30 \pm \sqrt{1764}}{2 \cdot 8}$

Step 7.1.4

Rewrite $1764$ as $42^{2}$ .

$x = \frac{- 30 \pm \sqrt{42^{2}}}{2 \cdot 8}$

Step 7.1.5

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

$x = \frac{- 30 \pm 42}{2 \cdot 8}$

Step 7.2

Multiply $2$ by $8$ .

$x = \frac{- 30 \pm 42}{16}$

Step 7.3

Simplify $\frac{- 30 \pm 42}{16}$ .

$x = \frac{- 15 \pm 21}{8}$

Step 7.4

Change the $\pm$ to $+$ .

$x = \frac{- 15 + 21}{8}$

Step 7.5

Add $- 15$ and $21$ .

$x = \frac{6}{8}$

Step 7.6

Cancel the common factor of $6$ and $8$ .

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Step 7.6.1

Factor $2$ out of $6$ .

$x = \frac{2 (3)}{8}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 3}{2 \cdot 4}$

Step 7.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot 3}{2 \cdot 4}$

Step 7.6.2.3

Rewrite the expression.

$x = \frac{3}{4}$

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 $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 8 \cdot - 27}}{2 \cdot 8}$

Step 8.1.2

Multiply $- 4 \cdot 8 \cdot - 27$ .

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Step 8.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{- 30 \pm \sqrt{900 - 32 \cdot - 27}}{2 \cdot 8}$

Step 8.1.2.2

Multiply $- 32$ by $- 27$ .

$x = \frac{- 30 \pm \sqrt{900 + 864}}{2 \cdot 8}$

Step 8.1.3

Add $900$ and $864$ .

$x = \frac{- 30 \pm \sqrt{1764}}{2 \cdot 8}$

Step 8.1.4

Rewrite $1764$ as $42^{2}$ .

$x = \frac{- 30 \pm \sqrt{42^{2}}}{2 \cdot 8}$

Step 8.1.5

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

$x = \frac{- 30 \pm 42}{2 \cdot 8}$

Step 8.2

Multiply $2$ by $8$ .

$x = \frac{- 30 \pm 42}{16}$

Step 8.3

Simplify $\frac{- 30 \pm 42}{16}$ .

$x = \frac{- 15 \pm 21}{8}$

Step 8.4

Change the $\pm$ to $-$ .

$x = \frac{- 15 - 21}{8}$

Step 8.5

Subtract $21$ from $- 15$ .

$x = \frac{- 36}{8}$

Step 8.6

Cancel the common factor of $- 36$ and $8$ .

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Step 8.6.1

Factor $4$ out of $- 36$ .

$x = \frac{4 (- 9)}{8}$

Step 8.6.2

Cancel the common factors.

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Step 8.6.2.1

Factor $4$ out of $8$ .

$x = \frac{4 \cdot - 9}{4 \cdot 2}$

Step 8.6.2.2

Cancel the common factor.

$x = \frac{4 \cdot - 9}{4 \cdot 2}$

Step 8.6.2.3

Rewrite the expression.

$x = \frac{- 9}{2}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{9}{2}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{3}{4}, - \frac{9}{2}$