Calculus Examples

$y - (- 109) = - 45 x (x - (- 5))$

Step 1

Rewrite the equation as $- 45 x (x - (- 5)) = y - (- 109)$ .

$- 45 x (x - (- 5)) = y - (- 109)$

Step 2

Simplify.

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Step 2.1

Multiply $- 1$ by $- 5$ .

$- 45 x (x + 5) = y - (- 109)$

Step 2.2

Multiply $- 1$ by $- 109$ .

$- 45 x (x + 5) = y + 109$

Step 3

Divide each term in $- 45 x (x + 5) = y + 109$ by $- 45$ and simplify.

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Step 3.1

Divide each term in $- 45 x (x + 5) = y + 109$ by $- 45$ .

$\frac{- 45 x (x + 5)}{- 45} = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 45$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 45 x (x + 5)}{- 45} = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.2.1.2

Divide $x (x + 5)$ by $1$ .

$x (x + 5) = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.2.2

Apply the distributive property.

$x \cdot x + x \cdot 5 = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 5 = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.2.3.2

Move $5$ to the left of $x$ .

$x^{2} + 5 x = \frac{y}{- 45} + \frac{109}{- 45}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$x^{2} + 5 x = - \frac{y}{45} + \frac{109}{- 45}$

Step 3.3.1.2

Move the negative in front of the fraction.

$x^{2} + 5 x = - \frac{y}{45} - \frac{109}{45}$

Step 4

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

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Step 4.1

Add $\frac{y}{45}$ to both sides of the equation.

$x^{2} + 5 x + \frac{y}{45} = - \frac{109}{45}$

Step 4.2

Add $\frac{109}{45}$ to both sides of the equation.

$x^{2} + 5 x + \frac{y}{45} + \frac{109}{45} = 0$

Step 5

Multiply through by the least common denominator $45$ , then simplify.

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Step 5.1

Apply the distributive property.

$45 x^{2} + 45 (5 x) + 45 (\frac{y}{45}) + 45 (\frac{109}{45}) = 0$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $5$ by $45$ .

$45 x^{2} + 225 x + 45 (\frac{y}{45}) + 45 (\frac{109}{45}) = 0$

Step 5.2.2

Cancel the common factor of $45$ .

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Step 5.2.2.1

Cancel the common factor.

$45 x^{2} + 225 x + 45 (\frac{y}{45}) + 45 (\frac{109}{45}) = 0$

Step 5.2.2.2

Rewrite the expression.

$45 x^{2} + 225 x + y + 45 (\frac{109}{45}) = 0$

Step 5.2.3

Cancel the common factor of $45$ .

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Step 5.2.3.1

Cancel the common factor.

$45 x^{2} + 225 x + y + 45 (\frac{109}{45}) = 0$

Step 5.2.3.2

Rewrite the expression.

$45 x^{2} + 225 x + y + 109 = 0$

Step 5.3

Move $225 x$ .

$45 x^{2} + y + 225 x + 109 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 45$ , $b = 225$ , and $c = y + 109$ into the quadratic formula and solve for $x$ .

$\frac{- 225 \pm \sqrt{225^{2} - 4 \cdot (45 \cdot (y + 109))}}{2 \cdot 45}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $225$ to the power of $2$ .

$x = \frac{- 225 \pm \sqrt{50625 - 4 \cdot 45 \cdot (y + 109)}}{2 \cdot 45}$

Step 8.1.2

Multiply $- 4$ by $45$ .

$x = \frac{- 225 \pm \sqrt{50625 - 180 \cdot (y + 109)}}{2 \cdot 45}$

Step 8.1.3

Apply the distributive property.

$x = \frac{- 225 \pm \sqrt{50625 - 180 y - 180 \cdot 109}}{2 \cdot 45}$

Step 8.1.4

Multiply $- 180$ by $109$ .

$x = \frac{- 225 \pm \sqrt{50625 - 180 y - 19620}}{2 \cdot 45}$

Step 8.1.5

Subtract $19620$ from $50625$ .

$x = \frac{- 225 \pm \sqrt{- 180 y + 31005}}{2 \cdot 45}$

Step 8.1.6

Factor $45$ out of $- 180 y + 31005$ .

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Step 8.1.6.1

Factor $45$ out of $- 180 y$ .

$x = \frac{- 225 \pm \sqrt{45 (- 4 y) + 31005}}{2 \cdot 45}$

Step 8.1.6.2

Factor $45$ out of $31005$ .

$x = \frac{- 225 \pm \sqrt{45 (- 4 y) + 45 (689)}}{2 \cdot 45}$

Step 8.1.6.3

Factor $45$ out of $45 (- 4 y) + 45 (689)$ .

$x = \frac{- 225 \pm \sqrt{45 (- 4 y + 689)}}{2 \cdot 45}$

Step 8.1.7

Rewrite $45 (- 4 y + 689)$ as $3^{2} \cdot (5 (- 4 y + 689))$ .

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Step 8.1.7.1

Factor $9$ out of $45$ .

$x = \frac{- 225 \pm \sqrt{9 (5) (- 4 y + 689)}}{2 \cdot 45}$

Step 8.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 225 \pm \sqrt{3^{2} \cdot (5 (- 4 y + 689))}}{2 \cdot 45}$

Step 8.1.7.3

Add parentheses.

$x = \frac{- 225 \pm \sqrt{3^{2} \cdot (5 (- 4 y + 689))}}{2 \cdot 45}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{- 225 \pm 3 \sqrt{5 (- 4 y + 689)}}{2 \cdot 45}$

Step 8.2

Multiply $2$ by $45$ .

$x = \frac{- 225 \pm 3 \sqrt{5 (- 4 y + 689)}}{90}$

Step 8.3

Simplify $\frac{- 225 \pm 3 \sqrt{5 (- 4 y + 689)}}{90}$ .

$x = \frac{- 75 \pm \sqrt{5 (- 4 y + 689)}}{30}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{75 - \sqrt{5 (- 4 y + 689)}}{30}$

$x = - \frac{75 + \sqrt{5 (- 4 y + 689)}}{30}$