Finite Math Examples

$y = 5 x^{2} + 15 x + 11$

Step 1

Rewrite the equation as $5 x^{2} + 15 x + 11 = y$ .

$5 x^{2} + 15 x + 11 = y$

Step 2

Subtract $y$ from both sides of the equation.

$5 x^{2} + 15 x + 11 - y = 0$

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 $a = 5$ , $b = 15$ , and $c = 11 - y$ into the quadratic formula and solve for $x$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (5 \cdot (11 - y))}}{2 \cdot 5}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot 5 \cdot (11 - y)}}{2 \cdot 5}$

Step 5.1.2

Multiply $- 4$ by $5$ .

$x = \frac{- 15 \pm \sqrt{225 - 20 \cdot (11 - y)}}{2 \cdot 5}$

Step 5.1.3

Apply the distributive property.

$x = \frac{- 15 \pm \sqrt{225 - 20 \cdot 11 - 20 (- y)}}{2 \cdot 5}$

Step 5.1.4

Multiply $- 20$ by $11$ .

$x = \frac{- 15 \pm \sqrt{225 - 220 - 20 (- y)}}{2 \cdot 5}$

Step 5.1.5

Multiply $- 1$ by $- 20$ .

$x = \frac{- 15 \pm \sqrt{225 - 220 + 20 y}}{2 \cdot 5}$

Step 5.1.6

Subtract $220$ from $225$ .

$x = \frac{- 15 \pm \sqrt{5 + 20 y}}{2 \cdot 5}$

Step 5.1.7

Factor $5$ out of $5 + 20 y$ .

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Step 5.1.7.1

Factor $5$ out of $5$ .

$x = \frac{- 15 \pm \sqrt{5 (1) + 20 y}}{2 \cdot 5}$

Step 5.1.7.2

Factor $5$ out of $20 y$ .

$x = \frac{- 15 \pm \sqrt{5 (1) + 5 (4 y)}}{2 \cdot 5}$

Step 5.1.7.3

Factor $5$ out of $5 (1) + 5 (4 y)$ .

$x = \frac{- 15 \pm \sqrt{5 (1 + 4 y)}}{2 \cdot 5}$

Step 5.2

Multiply $2$ by $5$ .

$x = \frac{- 15 \pm \sqrt{5 (1 + 4 y)}}{10}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{15 - \sqrt{5 (1 + 4 y)}}{10}$

$x = - \frac{15 + \sqrt{5 (4 y + 1)}}{10}$