Pre-Algebra Examples

$2 z^{2} - 2 z - \frac{143}{2} = 0$

Step 1

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

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Step 1.1

Apply the distributive property.

$2 (2 z^{2}) + 2 (- 2 z) + 2 (- \frac{143}{2}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $2$ by $2$ .

$4 z^{2} + 2 (- 2 z) + 2 (- \frac{143}{2}) = 0$

Step 1.2.2

Multiply $- 2$ by $2$ .

$4 z^{2} - 4 z + 2 (- \frac{143}{2}) = 0$

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Move the leading negative in $- \frac{143}{2}$ into the numerator.

$4 z^{2} - 4 z + 2 (\frac{- 143}{2}) = 0$

Step 1.2.3.2

Cancel the common factor.

$4 z^{2} - 4 z + 2 (\frac{- 143}{2}) = 0$

Step 1.2.3.3

Rewrite the expression.

$4 z^{2} - 4 z - 143 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 4$ , $b = - 4$ , and $c = - 143$ into the quadratic formula and solve for $z$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (4 \cdot - 143)}}{2 \cdot 4}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 4$ to the power of $2$ .

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 143}}{2 \cdot 4}$

Step 4.1.2

Multiply $- 4 \cdot 4 \cdot - 143$ .

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Step 4.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot - 143}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $- 143$ .

$z = \frac{4 \pm \sqrt{16 + 2288}}{2 \cdot 4}$

Step 4.1.3

Add $16$ and $2288$ .

$z = \frac{4 \pm \sqrt{2304}}{2 \cdot 4}$

Step 4.1.4

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

$z = \frac{4 \pm \sqrt{48^{2}}}{2 \cdot 4}$

Step 4.1.5

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

$z = \frac{4 \pm 48}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 48}{8}$

Step 4.3

Simplify $\frac{4 \pm 48}{8}$ .

$z = \frac{1 \pm 12}{2}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 143}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 143$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot - 143}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 143$ .

$z = \frac{4 \pm \sqrt{16 + 2288}}{2 \cdot 4}$

Step 5.1.3

Add $16$ and $2288$ .

$z = \frac{4 \pm \sqrt{2304}}{2 \cdot 4}$

Step 5.1.4

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

$z = \frac{4 \pm \sqrt{48^{2}}}{2 \cdot 4}$

Step 5.1.5

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

$z = \frac{4 \pm 48}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 48}{8}$

Step 5.3

Simplify $\frac{4 \pm 48}{8}$ .

$z = \frac{1 \pm 12}{2}$

Step 5.4

Change the $\pm$ to $+$ .

$z = \frac{1 + 12}{2}$

Step 5.5

Add $1$ and $12$ .

$z = \frac{13}{2}$

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 $- 4$ to the power of $2$ .

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot - 143}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 143$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot - 143}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 143$ .

$z = \frac{4 \pm \sqrt{16 + 2288}}{2 \cdot 4}$

Step 6.1.3

Add $16$ and $2288$ .

$z = \frac{4 \pm \sqrt{2304}}{2 \cdot 4}$

Step 6.1.4

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

$z = \frac{4 \pm \sqrt{48^{2}}}{2 \cdot 4}$

Step 6.1.5

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

$z = \frac{4 \pm 48}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 48}{8}$

Step 6.3

Simplify $\frac{4 \pm 48}{8}$ .

$z = \frac{1 \pm 12}{2}$

Step 6.4

Change the $\pm$ to $-$ .

$z = \frac{1 - 12}{2}$

Step 6.5

Subtract $12$ from $1$ .

$z = \frac{- 11}{2}$

Step 6.6

Move the negative in front of the fraction.

$z = - \frac{11}{2}$

Step 7

The final answer is the combination of both solutions.

$z = \frac{13}{2}, - \frac{11}{2}$