Finite Math Examples

$\frac{y^{2}}{2} - \frac{11}{6} y + 1 = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $y$ and $\frac{11}{6}$ .

$\frac{y^{2}}{2} - \frac{y \cdot 11}{6} + 1 = 0$

Step 1.1.2

Move $11$ to the left of $y$ .

$\frac{y^{2}}{2} - \frac{11 y}{6} + 1 = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

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Step 2.1

Apply the distributive property.

$6 (\frac{y^{2}}{2}) + 6 (- \frac{11 y}{6}) + 6 \cdot 1 = 0$

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

Factor $2$ out of $6$ .

$2 (3) (\frac{y^{2}}{2}) + 6 (- \frac{11 y}{6}) + 6 \cdot 1 = 0$

Step 2.2.1.2

Cancel the common factor.

$2 \cdot (3 (\frac{y^{2}}{2})) + 6 (- \frac{11 y}{6}) + 6 \cdot 1 = 0$

Step 2.2.1.3

Rewrite the expression.

$3 y^{2} + 6 (- \frac{11 y}{6}) + 6 \cdot 1 = 0$

Step 2.2.2

Cancel the common factor of $6$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{11 y}{6}$ into the numerator.

$3 y^{2} + 6 (\frac{- 11 y}{6}) + 6 \cdot 1 = 0$

Step 2.2.2.2

Cancel the common factor.

$3 y^{2} + 6 (\frac{- 11 y}{6}) + 6 \cdot 1 = 0$

Step 2.2.2.3

Rewrite the expression.

$3 y^{2} - 11 y + 6 \cdot 1 = 0$

Step 2.2.3

Multiply $6$ by $1$ .

$3 y^{2} - 11 y + 6 = 0$

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 $a = 3$ , $b = - 11$ , and $c = 6$ into the quadratic formula and solve for $y$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (3 \cdot 6)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 11$ to the power of $2$ .

$y = \frac{11 \pm \sqrt{121 - 4 \cdot 3 \cdot 6}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 6$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{11 \pm \sqrt{121 - 12 \cdot 6}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $6$ .

$y = \frac{11 \pm \sqrt{121 - 72}}{2 \cdot 3}$

Step 5.1.3

Subtract $72$ from $121$ .

$y = \frac{11 \pm \sqrt{49}}{2 \cdot 3}$

Step 5.1.4

Rewrite $49$ as $7^{2}$ .

$y = \frac{11 \pm \sqrt{7^{2}}}{2 \cdot 3}$

Step 5.1.5

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

$y = \frac{11 \pm 7}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$y = \frac{11 \pm 7}{6}$

Step 6

The final answer is the combination of both solutions.

$y = 3, \frac{2}{3}$