Algebra Examples

$z (x) = 16 x^{2} + 8 x + 1$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$16 x^{2} + 8 x + 1 = z x$

Step 2

Subtract $z x$ from both sides of the equation.

$16 x^{2} + 8 x + 1 - z x = 0$

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 $a = 16$ , $b = 8 - z$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- (8 - z) \pm \sqrt{{(8 - z)}^{2} - 4 \cdot (16 \cdot 1)}}{2 \cdot 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$x = \frac{- 1 \cdot 8 + z \pm \sqrt{{(8 - z)}^{2} - 4 \cdot 16 \cdot 1}}{2 \cdot 16}$

Step 5.1.2

Multiply $- 1$ by $8$ .

$x = \frac{- 8 + z \pm \sqrt{{(8 - z)}^{2} - 4 \cdot 16 \cdot 1}}{2 \cdot 16}$

Step 5.1.3

Multiply $- - z$ .

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Step 5.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- 8 + 1 z \pm \sqrt{{(8 - z)}^{2} - 4 \cdot 16 \cdot 1}}{2 \cdot 16}$

Step 5.1.3.2

Multiply $z$ by $1$ .

$x = \frac{- 8 + z \pm \sqrt{{(8 - z)}^{2} - 4 \cdot 16 \cdot 1}}{2 \cdot 16}$

Step 5.1.4

Rewrite $4 \cdot 16 \cdot 1$ as ${(2 \cdot 4 \cdot 1)}^{2}$ .

$x = \frac{- 8 + z \pm \sqrt{{(8 - z)}^{2} - {(2 \cdot 4 \cdot 1)}^{2}}}{2 \cdot 16}$

Step 5.1.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8 - z$ and $b = 2 \cdot 4 \cdot 1$ .

$x = \frac{- 8 + z \pm \sqrt{(8 - z + 2 \cdot 4 \cdot 1) (8 - z - (2 \cdot 4 \cdot 1))}}{2 \cdot 16}$

Step 5.1.6

Simplify.

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Step 5.1.6.1

Multiply $2 \cdot 4 \cdot 1$ .

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Step 5.1.6.1.1

Multiply $2$ by $4$ .

$x = \frac{- 8 + z \pm \sqrt{(8 - z + 8 \cdot 1) (8 - z - (2 \cdot 4 \cdot 1))}}{2 \cdot 16}$

Step 5.1.6.1.2

Multiply $8$ by $1$ .

$x = \frac{- 8 + z \pm \sqrt{(8 - z + 8) (8 - z - (2 \cdot 4 \cdot 1))}}{2 \cdot 16}$

Step 5.1.6.2

Add $8$ and $8$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) (8 - z - (2 \cdot 4 \cdot 1))}}{2 \cdot 16}$

Step 5.1.6.3

Multiply $- (2 \cdot 4 \cdot 1)$ .

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Step 5.1.6.3.1

Multiply $2$ by $4$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) (8 - z - (8 \cdot 1))}}{2 \cdot 16}$

Step 5.1.6.3.2

Multiply $8$ by $1$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) (8 - z - 1 \cdot 8)}}{2 \cdot 16}$

Step 5.1.6.3.3

Multiply $- 1$ by $8$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) (8 - z - 8)}}{2 \cdot 16}$

Step 5.1.6.4

Subtract $8$ from $8$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) (- z + 0)}}{2 \cdot 16}$

Step 5.1.6.5

Add $- z$ and $0$ .

$x = \frac{- 8 + z \pm \sqrt{(- z + 16) \cdot (- 1 z)}}{2 \cdot 16}$

Step 5.1.6.6

Factor out negative.

$x = \frac{- 8 + z \pm \sqrt{- ((- z + 16) z)}}{2 \cdot 16}$

$x = \frac{- 8 + z \pm \sqrt{- (- z + 16) z}}{2 \cdot 16}$

Step 5.2

Multiply $2$ by $16$ .

$x = \frac{- 8 + z \pm \sqrt{- (- z + 16) z}}{32}$

Step 5.3

Reorder factors in $\frac{- 8 + z \pm \sqrt{- (- z + 16) z}}{32}$ .

$x = \frac{- 8 + z \pm \sqrt{- z (- z + 16)}}{32}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{8 - z - \sqrt{- z (- z + 16)}}{32}$

$x = - \frac{8 - z + \sqrt{- z (- z + 16)}}{32}$