Precalculus Examples

$z^{2} - \frac{3}{2} z + \frac{9}{16} = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$z^{2} - \frac{z \cdot 3}{2} + \frac{9}{16} = 0$

Step 1.1.2

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

$z^{2} - \frac{3 z}{2} + \frac{9}{16} = 0$

Step 2

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

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Step 2.1

Apply the distributive property.

$16 z^{2} + 16 (- \frac{3 z}{2}) + 16 (\frac{9}{16}) = 0$

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 $- \frac{3 z}{2}$ into the numerator.

$16 z^{2} + 16 (\frac{- 3 z}{2}) + 16 (\frac{9}{16}) = 0$

Step 2.2.1.2

Factor $2$ out of $16$ .

$16 z^{2} + 2 (8) (\frac{- 3 z}{2}) + 16 (\frac{9}{16}) = 0$

Step 2.2.1.3

Cancel the common factor.

$16 z^{2} + 2 \cdot (8 (\frac{- 3 z}{2})) + 16 (\frac{9}{16}) = 0$

Step 2.2.1.4

Rewrite the expression.

$16 z^{2} + 8 (- 3 z) + 16 (\frac{9}{16}) = 0$

Step 2.2.2

Multiply $- 3$ by $8$ .

$16 z^{2} - 24 z + 16 (\frac{9}{16}) = 0$

Step 2.2.3

Cancel the common factor of $16$ .

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Step 2.2.3.1

Cancel the common factor.

$16 z^{2} - 24 z + 16 (\frac{9}{16}) = 0$

Step 2.2.3.2

Rewrite the expression.

$16 z^{2} - 24 z + 9 = 0$

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 $c = 9$ into the quadratic formula and solve for $z$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (16 \cdot 9)}}{2 \cdot 16}$

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 $2$ .

$z = \frac{24 \pm \sqrt{576 - 4 \cdot 16 \cdot 9}}{2 \cdot 16}$

Step 5.1.2

Multiply $- 4 \cdot 16 \cdot 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $16$ .

$z = \frac{24 \pm \sqrt{576 - 64 \cdot 9}}{2 \cdot 16}$

Step 5.1.2.2

Multiply $- 64$ by $9$ .

$z = \frac{24 \pm \sqrt{576 - 576}}{2 \cdot 16}$

Step 5.1.3

Subtract $576$ from $576$ .

$z = \frac{24 \pm \sqrt{0}}{2 \cdot 16}$

Step 5.1.4

Rewrite $0$ as $0^{2}$ .

$z = \frac{24 \pm \sqrt{0^{2}}}{2 \cdot 16}$

Step 5.1.5

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

$z = \frac{24 \pm 0}{2 \cdot 16}$

Step 5.1.6

$24$ plus or minus $0$ is $24$ .

$z = \frac{24}{2 \cdot 16}$

Step 5.2

Multiply $2$ by $16$ .

$z = \frac{24}{32}$

Step 5.3

Cancel the common factor of $24$ and $32$ .

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Step 5.3.1

Factor $8$ out of $24$ .

$z = \frac{8 (3)}{32}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $8$ out of $32$ .

$z = \frac{8 \cdot 3}{8 \cdot 4}$

Step 5.3.2.2

Cancel the common factor.

$z = \frac{8 \cdot 3}{8 \cdot 4}$

Step 5.3.2.3

Rewrite the expression.

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

Step 6

The final answer is the combination of both solutions.

$z = \frac{3}{4}$ Double roots