Finite Math Examples

$y = y^{2} - 10$

Step 1

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

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Step 1.1

Subtract $y^{2}$ from both sides of the equation.

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

Step 1.2

Add $10$ to both sides of the equation.

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

Step 2

Reorder terms.

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

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

One to any power is one.

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

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 5.1.2.2

Multiply $4$ by $10$ .

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

Step 5.1.3

Add $1$ and $40$ .

$y = \frac{- 1 \pm \sqrt{41}}{2 \cdot - 1}$

Step 5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 1 \pm \sqrt{41}}{- 2}$

Step 5.3

Simplify $\frac{- 1 \pm \sqrt{41}}{- 2}$ .

$y = \frac{1 \pm \sqrt{41}}{2}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

One to any power is one.

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

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 6.1.2.2

Multiply $4$ by $10$ .

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

Step 6.1.3

Add $1$ and $40$ .

$y = \frac{- 1 \pm \sqrt{41}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 1 \pm \sqrt{41}}{- 2}$

Step 6.3

Simplify $\frac{- 1 \pm \sqrt{41}}{- 2}$ .

$y = \frac{1 \pm \sqrt{41}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$y = \frac{1 + \sqrt{41}}{2}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

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

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 7.1.2.2

Multiply $4$ by $10$ .

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

Step 7.1.3

Add $1$ and $40$ .

$y = \frac{- 1 \pm \sqrt{41}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$y = \frac{- 1 \pm \sqrt{41}}{- 2}$

Step 7.3

Simplify $\frac{- 1 \pm \sqrt{41}}{- 2}$ .

$y = \frac{1 \pm \sqrt{41}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$y = \frac{1 - \sqrt{41}}{2}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{41}}{2}, \frac{1 - \sqrt{41}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$y = \frac{1 + \sqrt{41}}{2}, \frac{1 - \sqrt{41}}{2}$

Decimal Form:

$y = 3.70156211 \dots, - 2.70156211 \dots$