Basic Math Examples

$\frac{1}{2} y^{2} - y + 5 = 0$

Step 1

Combine $\frac{1}{2}$ and $y^{2}$ .

$\frac{y^{2}}{2} - y + 5 = 0$

Step 2

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

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Step 2.1

Apply the distributive property.

$2 (\frac{y^{2}}{2}) + 2 (- y) + 2 \cdot 5 = 0$

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

Cancel the common factor.

$2 (\frac{y^{2}}{2}) + 2 (- y) + 2 \cdot 5 = 0$

Step 2.2.1.2

Rewrite the expression.

$y^{2} + 2 (- y) + 2 \cdot 5 = 0$

Step 2.2.2

Multiply $- 1$ by $2$ .

$y^{2} - 2 y + 2 \cdot 5 = 0$

Step 2.2.3

Multiply $2$ by $5$ .

$y^{2} - 2 y + 10 = 0$

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 $a = 1$ , $b = - 2$ , and $c = 10$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 10}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $10$ .

$y = \frac{2 \pm \sqrt{4 - 40}}{2 \cdot 1}$

Step 5.1.3

Subtract $40$ from $4$ .

$y = \frac{2 \pm \sqrt{- 36}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 36}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot 1}$

Step 5.1.6

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

$y = \frac{2 \pm i \cdot \sqrt{36}}{2 \cdot 1}$

Step 5.1.7

Rewrite $36$ as $6^{2}$ .

$y = \frac{2 \pm i \cdot \sqrt{6^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$y = \frac{2 \pm i \cdot 6}{2 \cdot 1}$

Step 5.1.9

Move $6$ to the left of $i$ .

$y = \frac{2 \pm 6 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$y = \frac{2 \pm 6 i}{2}$

Step 5.3

Simplify $\frac{2 \pm 6 i}{2}$ .

$y = 1 \pm 3 i$

Step 6

The final answer is the combination of both solutions.

$y = 1 + 3 i, 1 - 3 i$