Pre-Algebra Examples

$x^{2} - \frac{3}{2} x - \frac{27}{16} = 0$

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{3}{2}$ .

$x^{2} - \frac{x \cdot 3}{2} - \frac{27}{16} = 0$

Step 1.2

Move $3$ to the left of $x$ .

$x^{2} - \frac{3 x}{2} - \frac{27}{16} = 0$

Step 2

Multiply through by the least common denominator $16$ , then simplify.

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Step 2.1

Apply the distributive property.

$16 x^{2} + 16 (- \frac{3 x}{2}) + 16 (- \frac{27}{16}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Move the leading negative in $- \frac{3 x}{2}$ into the numerator.

$16 x^{2} + 16 (\frac{- 3 x}{2}) + 16 (- \frac{27}{16}) = 0$

Step 2.2.1.2

Factor $2$ out of $16$ .

$16 x^{2} + 2 (8) (\frac{- 3 x}{2}) + 16 (- \frac{27}{16}) = 0$

Step 2.2.1.3

Cancel the common factor.

$16 x^{2} + 2 \cdot (8 (\frac{- 3 x}{2})) + 16 (- \frac{27}{16}) = 0$

Step 2.2.1.4

Rewrite the expression.

$16 x^{2} + 8 (- 3 x) + 16 (- \frac{27}{16}) = 0$

Step 2.2.2

Multiply $- 3$ by $8$ .

$16 x^{2} - 24 x + 16 (- \frac{27}{16}) = 0$

Step 2.2.3

Cancel the common factor of $16$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{27}{16}$ into the numerator.

$16 x^{2} - 24 x + 16 (\frac{- 27}{16}) = 0$

Step 2.2.3.2

Cancel the common factor.

$16 x^{2} - 24 x + 16 (\frac{- 27}{16}) = 0$

Step 2.2.3.3

Rewrite the expression.

$16 x^{2} - 24 x - 27 = 0$

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 $a = 16$ , $b = - 24$ , and $c = - 27$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (16 \cdot - 27)}}{2 \cdot 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 16 \cdot - 27}}{2 \cdot 16}$

Step 5.1.2

Multiply $- 4 \cdot 16 \cdot - 27$ .

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Step 5.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{24 \pm \sqrt{576 - 64 \cdot - 27}}{2 \cdot 16}$

Step 5.1.2.2

Multiply $- 64$ by $- 27$ .

$x = \frac{24 \pm \sqrt{576 + 1728}}{2 \cdot 16}$

Step 5.1.3

Add $576$ and $1728$ .

$x = \frac{24 \pm \sqrt{2304}}{2 \cdot 16}$

Step 5.1.4

Rewrite $2304$ as $48^{2}$ .

$x = \frac{24 \pm \sqrt{48^{2}}}{2 \cdot 16}$

Step 5.1.5

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

$x = \frac{24 \pm 48}{2 \cdot 16}$

Step 5.2

Multiply $2$ by $16$ .

$x = \frac{24 \pm 48}{32}$

Step 5.3

Simplify $\frac{24 \pm 48}{32}$ .

$x = \frac{3 \pm 6}{4}$

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 $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 16 \cdot - 27}}{2 \cdot 16}$

Step 6.1.2

Multiply $- 4 \cdot 16 \cdot - 27$ .

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Step 6.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{24 \pm \sqrt{576 - 64 \cdot - 27}}{2 \cdot 16}$

Step 6.1.2.2

Multiply $- 64$ by $- 27$ .

$x = \frac{24 \pm \sqrt{576 + 1728}}{2 \cdot 16}$

Step 6.1.3

Add $576$ and $1728$ .

$x = \frac{24 \pm \sqrt{2304}}{2 \cdot 16}$

Step 6.1.4

Rewrite $2304$ as $48^{2}$ .

$x = \frac{24 \pm \sqrt{48^{2}}}{2 \cdot 16}$

Step 6.1.5

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

$x = \frac{24 \pm 48}{2 \cdot 16}$

Step 6.2

Multiply $2$ by $16$ .

$x = \frac{24 \pm 48}{32}$

Step 6.3

Simplify $\frac{24 \pm 48}{32}$ .

$x = \frac{3 \pm 6}{4}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{3 + 6}{4}$

Step 6.5

Add $3$ and $6$ .

$x = \frac{9}{4}$

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 $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 16 \cdot - 27}}{2 \cdot 16}$

Step 7.1.2

Multiply $- 4 \cdot 16 \cdot - 27$ .

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Step 7.1.2.1

Multiply $- 4$ by $16$ .

$x = \frac{24 \pm \sqrt{576 - 64 \cdot - 27}}{2 \cdot 16}$

Step 7.1.2.2

Multiply $- 64$ by $- 27$ .

$x = \frac{24 \pm \sqrt{576 + 1728}}{2 \cdot 16}$

Step 7.1.3

Add $576$ and $1728$ .

$x = \frac{24 \pm \sqrt{2304}}{2 \cdot 16}$

Step 7.1.4

Rewrite $2304$ as $48^{2}$ .

$x = \frac{24 \pm \sqrt{48^{2}}}{2 \cdot 16}$

Step 7.1.5

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

$x = \frac{24 \pm 48}{2 \cdot 16}$

Step 7.2

Multiply $2$ by $16$ .

$x = \frac{24 \pm 48}{32}$

Step 7.3

Simplify $\frac{24 \pm 48}{32}$ .

$x = \frac{3 \pm 6}{4}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{3 - 6}{4}$

Step 7.5

Subtract $6$ from $3$ .

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

Step 7.6

Move the negative in front of the fraction.

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

Step 8

The final answer is the combination of both solutions.

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