Basic Math Examples

$72 y^{2} + 1 = 12 y$

Step 1

Subtract $12 y$ from both sides of the equation.

$72 y^{2} + 1 - 12 y = 0$

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 $a = 72$ , $b = - 12$ , and $c = 1$ into the quadratic formula and solve for $y$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (72 \cdot 1)}}{2 \cdot 72}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 12$ to the power of $2$ .

$y = \frac{12 \pm \sqrt{144 - 4 \cdot 72 \cdot 1}}{2 \cdot 72}$

Step 4.1.2

Multiply $- 4 \cdot 72 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $72$ .

$y = \frac{12 \pm \sqrt{144 - 288 \cdot 1}}{2 \cdot 72}$

Step 4.1.2.2

Multiply $- 288$ by $1$ .

$y = \frac{12 \pm \sqrt{144 - 288}}{2 \cdot 72}$

Step 4.1.3

Subtract $288$ from $144$ .

$y = \frac{12 \pm \sqrt{- 144}}{2 \cdot 72}$

Step 4.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$y = \frac{12 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 72}$

Step 4.1.5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$y = \frac{12 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 72}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{12 \pm i \cdot \sqrt{144}}{2 \cdot 72}$

Step 4.1.7

Rewrite $144$ as $12^{2}$ .

$y = \frac{12 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 72}$

Step 4.1.8

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

$y = \frac{12 \pm i \cdot 12}{2 \cdot 72}$

Step 4.1.9

Move $12$ to the left of $i$ .

$y = \frac{12 \pm 12 i}{2 \cdot 72}$

Step 4.2

Multiply $2$ by $72$ .

$y = \frac{12 \pm 12 i}{144}$

Step 4.3

Simplify $\frac{12 \pm 12 i}{144}$ .

$y = \frac{1 \pm i}{12}$

Step 5

The final answer is the combination of both solutions.

$y = \frac{1}{12} + \frac{i}{12}, \frac{1}{12} - \frac{i}{12}$