Basic Math Examples

$9 z^{2} - {(z + 1)}^{2} = y - 3$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $y$ from both sides of the equation.

$9 z^{2} - {(z + 1)}^{2} - y = - 3$

Step 1.1.2

Add $3$ to both sides of the equation.

$9 z^{2} - {(z + 1)}^{2} - y + 3 = 0$

Step 1.2

Simplify $9 z^{2} - {(z + 1)}^{2} - y + 3$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(z + 1)}^{2}$ as $(z + 1) (z + 1)$ .

$9 z^{2} - ((z + 1) (z + 1)) - y + 3 = 0$

Step 1.2.1.2

Expand $(z + 1) (z + 1)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$9 z^{2} - (z (z + 1) + 1 (z + 1)) - y + 3 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$9 z^{2} - (z \cdot z + z \cdot 1 + 1 (z + 1)) - y + 3 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$9 z^{2} - (z \cdot z + z \cdot 1 + 1 z + 1 \cdot 1) - y + 3 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $z$ by $z$ .

$9 z^{2} - (z^{2} + z \cdot 1 + 1 z + 1 \cdot 1) - y + 3 = 0$

Step 1.2.1.3.1.2

Multiply $z$ by $1$ .

$9 z^{2} - (z^{2} + z + 1 z + 1 \cdot 1) - y + 3 = 0$

Step 1.2.1.3.1.3

Multiply $z$ by $1$ .

$9 z^{2} - (z^{2} + z + z + 1 \cdot 1) - y + 3 = 0$

Step 1.2.1.3.1.4

Multiply $1$ by $1$ .

$9 z^{2} - (z^{2} + z + z + 1) - y + 3 = 0$

Step 1.2.1.3.2

Add $z$ and $z$ .

$9 z^{2} - (z^{2} + 2 z + 1) - y + 3 = 0$

Step 1.2.1.4

Apply the distributive property.

$9 z^{2} - z^{2} - (2 z) - 1 \cdot 1 - y + 3 = 0$

Step 1.2.1.5

Simplify.

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Step 1.2.1.5.1

Multiply $2$ by $- 1$ .

$9 z^{2} - z^{2} - 2 z - 1 \cdot 1 - y + 3 = 0$

Step 1.2.1.5.2

Multiply $- 1$ by $1$ .

$9 z^{2} - z^{2} - 2 z - 1 - y + 3 = 0$

Step 1.2.2

Subtract $z^{2}$ from $9 z^{2}$ .

$8 z^{2} - 2 z - 1 - y + 3 = 0$

Step 1.2.3

Add $- 1$ and $3$ .

$8 z^{2} - 2 z - y + 2 = 0$

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 $a = 8$ , $b = - 2$ , and $c = - y + 2$ into the quadratic formula and solve for $z$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (8 \cdot (- y + 2))}}{2 \cdot 8}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

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

Step 4.1.2

Multiply $- 4$ by $8$ .

$z = \frac{2 \pm \sqrt{4 - 32 \cdot (- y + 2)}}{2 \cdot 8}$

Step 4.1.3

Apply the distributive property.

$z = \frac{2 \pm \sqrt{4 - 32 (- y) - 32 \cdot 2}}{2 \cdot 8}$

Step 4.1.4

Multiply $- 1$ by $- 32$ .

$z = \frac{2 \pm \sqrt{4 + 32 y - 32 \cdot 2}}{2 \cdot 8}$

Step 4.1.5

Multiply $- 32$ by $2$ .

$z = \frac{2 \pm \sqrt{4 + 32 y - 64}}{2 \cdot 8}$

Step 4.1.6

Subtract $64$ from $4$ .

$z = \frac{2 \pm \sqrt{32 y - 60}}{2 \cdot 8}$

Step 4.1.7

Factor $4$ out of $32 y - 60$ .

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Step 4.1.7.1

Factor $4$ out of $32 y$ .

$z = \frac{2 \pm \sqrt{4 (8 y) - 60}}{2 \cdot 8}$

Step 4.1.7.2

Factor $4$ out of $- 60$ .

$z = \frac{2 \pm \sqrt{4 (8 y) + 4 (- 15)}}{2 \cdot 8}$

Step 4.1.7.3

Factor $4$ out of $4 (8 y) + 4 (- 15)$ .

$z = \frac{2 \pm \sqrt{4 (8 y - 15)}}{2 \cdot 8}$

Step 4.1.8

Rewrite $4$ as $2^{2}$ .

$z = \frac{2 \pm \sqrt{2^{2} (8 y - 15)}}{2 \cdot 8}$

Step 4.1.9

Pull terms out from under the radical.

$z = \frac{2 \pm 2 \sqrt{8 y - 15}}{2 \cdot 8}$

Step 4.2

Multiply $2$ by $8$ .

$z = \frac{2 \pm 2 \sqrt{8 y - 15}}{16}$

Step 4.3

Simplify $\frac{2 \pm 2 \sqrt{8 y - 15}}{16}$ .

$z = \frac{1 \pm \sqrt{8 y - 15}}{8}$

Step 5

The final answer is the combination of both solutions.

$z = \frac{1 + \sqrt{8 y - 15}}{8}$

$z = \frac{1 - \sqrt{8 y - 15}}{8}$