Basic Math Examples

$3 y^{2} - 2 y + 7 = 0 {(y - \frac{1}{3})}^{2}$

Step 1

Simplify $0 {(y - \frac{1}{3})}^{2}$ .

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Step 1.1

Rewrite.

$3 y^{2} - 2 y + 7 = 0 + 0 + 0 {(y - \frac{1}{3})}^{2}$

Step 1.2

Rewrite ${(y - \frac{1}{3})}^{2}$ as $(y - \frac{1}{3}) (y - \frac{1}{3})$ .

$3 y^{2} - 2 y + 7 = 0 ((y - \frac{1}{3}) (y - \frac{1}{3}))$

Step 1.3

Expand $(y - \frac{1}{3}) (y - \frac{1}{3})$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 y^{2} - 2 y + 7 = 0 (y (y - \frac{1}{3}) - \frac{1}{3} (y - \frac{1}{3}))$

Step 1.3.2

Apply the distributive property.

$3 y^{2} - 2 y + 7 = 0 (y \cdot y + y (- \frac{1}{3}) - \frac{1}{3} (y - \frac{1}{3}))$

Step 1.3.3

Apply the distributive property.

$3 y^{2} - 2 y + 7 = 0 (y \cdot y + y (- \frac{1}{3}) - \frac{1}{3} y - \frac{1}{3} (- \frac{1}{3}))$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $y$ by $y$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} + y (- \frac{1}{3}) - \frac{1}{3} y - \frac{1}{3} (- \frac{1}{3}))$

Step 1.4.1.2

Combine $y$ and $\frac{1}{3}$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{1}{3} y - \frac{1}{3} (- \frac{1}{3}))$

Step 1.4.1.3

Combine $y$ and $\frac{1}{3}$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{y}{3} - \frac{1}{3} (- \frac{1}{3}))$

Step 1.4.1.4

Multiply $- \frac{1}{3} (- \frac{1}{3})$ .

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Step 1.4.1.4.1

Multiply $- 1$ by $- 1$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{y}{3} + 1 (\frac{1}{3}) \frac{1}{3})$

Step 1.4.1.4.2

Multiply $\frac{1}{3}$ by $1$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{y}{3} + \frac{1}{3} \cdot \frac{1}{3})$

Step 1.4.1.4.3

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{y}{3} + \frac{1}{3 \cdot 3})$

Step 1.4.1.4.4

Multiply $3$ by $3$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{y}{3} - \frac{y}{3} + \frac{1}{9})$

Step 1.4.2

Subtract $\frac{y}{3}$ from $- \frac{y}{3}$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} - 2 \frac{y}{3} + \frac{1}{9})$

Step 1.5

Simplify each term.

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Step 1.5.1

Combine $- 2$ and $\frac{y}{3}$ .

$3 y^{2} - 2 y + 7 = 0 (y^{2} + \frac{- 2 y}{3} + \frac{1}{9})$

Step 1.5.2

Move the negative in front of the fraction.

$3 y^{2} - 2 y + 7 = 0 (y^{2} - \frac{2 y}{3} + \frac{1}{9})$

Step 1.6

Multiply $0$ by $y^{2} - \frac{2 y}{3} + \frac{1}{9}$ .

$3 y^{2} - 2 y + 7 = 0$

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 $a = 3$ , $b = - 2$ , and $c = 7$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot 7)}}{2 \cdot 3}$

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 $2$ .

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot 7}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{2 \pm \sqrt{4 - 12 \cdot 7}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $7$ .

$y = \frac{2 \pm \sqrt{4 - 84}}{2 \cdot 3}$

Step 4.1.3

Subtract $84$ from $4$ .

$y = \frac{2 \pm \sqrt{- 80}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 80$ as $- 1 (80)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (80)}$ as $\sqrt{- 1} \cdot \sqrt{80}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 3}$

Step 4.1.6

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

$y = \frac{2 \pm i \cdot \sqrt{80}}{2 \cdot 3}$

Step 4.1.7

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 4.1.7.1

Factor $16$ out of $80$ .

$y = \frac{2 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 3}$

Step 4.1.7.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{2 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 3}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{2 \pm i \cdot (4 \sqrt{5})}{2 \cdot 3}$

Step 4.1.9

Move $4$ to the left of $i$ .

$y = \frac{2 \pm 4 i \sqrt{5}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$y = \frac{2 \pm 4 i \sqrt{5}}{6}$

Step 4.3

Simplify $\frac{2 \pm 4 i \sqrt{5}}{6}$ .

$y = \frac{1 \pm 2 i \sqrt{5}}{3}$

Step 5

The final answer is the combination of both solutions.

$y = \frac{1 + 2 i \sqrt{5}}{3}, \frac{1 - 2 i \sqrt{5}}{3}$