Precalculus Examples

$x = 20 y^{2} - 120 y + 60$

Step 1

Solve for $y$ .

Tap for more steps...

Step 1.1

Rewrite the equation as $20 y^{2} - 120 y + 60 = x$ .

$20 y^{2} - 120 y + 60 = x$

Step 1.2

Subtract $x$ from both sides of the equation.

$20 y^{2} - 120 y + 60 - x = 0$

Step 1.3

Use the quadratic formula to find the solutions.

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

Step 1.4

Substitute the values $a = 20$ , $b = - 120$ , and $c = 60 - x$ into the quadratic formula and solve for $y$ .

$\frac{120 \pm \sqrt{{(- 120)}^{2} - 4 \cdot (20 \cdot (60 - x))}}{2 \cdot 20}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify the numerator.

Tap for more steps...

Step 1.5.1.1

Raise $- 120$ to the power of $2$ .

$y = \frac{120 \pm \sqrt{14400 - 4 \cdot 20 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.5.1.2

Multiply $- 4$ by $20$ .

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot 60 - 80 (- x)}}{2 \cdot 20}$

Step 1.5.1.4

Multiply $- 80$ by $60$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 - 80 (- x)}}{2 \cdot 20}$

Step 1.5.1.5

Multiply $- 1$ by $- 80$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 + 80 x}}{2 \cdot 20}$

Step 1.5.1.6

Subtract $4800$ from $14400$ .

$y = \frac{120 \pm \sqrt{9600 + 80 x}}{2 \cdot 20}$

Step 1.5.1.7

Factor $80$ out of $9600 + 80 x$ .

Tap for more steps...

Step 1.5.1.7.1

Factor $80$ out of $9600$ .

$y = \frac{120 \pm \sqrt{80 \cdot 120 + 80 x}}{2 \cdot 20}$

Step 1.5.1.7.2

Factor $80$ out of $80 \cdot 120 + 80 x$ .

$y = \frac{120 \pm \sqrt{80 (120 + x)}}{2 \cdot 20}$

Step 1.5.1.8

Rewrite $80 (120 + x)$ as ${(2^{2})}^{2} \cdot (5 (120 + x))$ .

Tap for more steps...

Step 1.5.1.8.1

Factor $16$ out of $80$ .

$y = \frac{120 \pm \sqrt{16 (5) (120 + x)}}{2 \cdot 20}$

Step 1.5.1.8.2

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

$y = \frac{120 \pm \sqrt{4^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.5.1.8.3

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

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.5.1.8.4

Add parentheses.

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.5.1.9

Pull terms out from under the radical.

$y = \frac{120 \pm 2^{2} \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.5.1.10

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

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.5.2

Multiply $2$ by $20$ .

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$

Step 1.5.3

Simplify $\frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$ .

$y = \frac{30 \pm \sqrt{5 (120 + x)}}{10}$

Step 1.6

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

Tap for more steps...

Step 1.6.1

Simplify the numerator.

Tap for more steps...

Step 1.6.1.1

Raise $- 120$ to the power of $2$ .

$y = \frac{120 \pm \sqrt{14400 - 4 \cdot 20 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.6.1.2

Multiply $- 4$ by $20$ .

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot 60 - 80 (- x)}}{2 \cdot 20}$

Step 1.6.1.4

Multiply $- 80$ by $60$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 - 80 (- x)}}{2 \cdot 20}$

Step 1.6.1.5

Multiply $- 1$ by $- 80$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 + 80 x}}{2 \cdot 20}$

Step 1.6.1.6

Subtract $4800$ from $14400$ .

$y = \frac{120 \pm \sqrt{9600 + 80 x}}{2 \cdot 20}$

Step 1.6.1.7

Factor $80$ out of $9600 + 80 x$ .

Tap for more steps...

Step 1.6.1.7.1

Factor $80$ out of $9600$ .

$y = \frac{120 \pm \sqrt{80 \cdot 120 + 80 x}}{2 \cdot 20}$

Step 1.6.1.7.2

Factor $80$ out of $80 \cdot 120 + 80 x$ .

$y = \frac{120 \pm \sqrt{80 (120 + x)}}{2 \cdot 20}$

Step 1.6.1.8

Rewrite $80 (120 + x)$ as ${(2^{2})}^{2} \cdot (5 (120 + x))$ .

Tap for more steps...

Step 1.6.1.8.1

Factor $16$ out of $80$ .

$y = \frac{120 \pm \sqrt{16 (5) (120 + x)}}{2 \cdot 20}$

Step 1.6.1.8.2

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

$y = \frac{120 \pm \sqrt{4^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.6.1.8.3

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

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.6.1.8.4

Add parentheses.

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.6.1.9

Pull terms out from under the radical.

$y = \frac{120 \pm 2^{2} \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.6.1.10

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

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.6.2

Multiply $2$ by $20$ .

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$

Step 1.6.3

Simplify $\frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$ .

$y = \frac{30 \pm \sqrt{5 (120 + x)}}{10}$

Step 1.6.4

Change the $\pm$ to $+$ .

$y = \frac{30 + \sqrt{5 (120 + x)}}{10}$

Step 1.7

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

Tap for more steps...

Step 1.7.1

Simplify the numerator.

Tap for more steps...

Step 1.7.1.1

Raise $- 120$ to the power of $2$ .

$y = \frac{120 \pm \sqrt{14400 - 4 \cdot 20 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.7.1.2

Multiply $- 4$ by $20$ .

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot (60 - x)}}{2 \cdot 20}$

Step 1.7.1.3

Apply the distributive property.

$y = \frac{120 \pm \sqrt{14400 - 80 \cdot 60 - 80 (- x)}}{2 \cdot 20}$

Step 1.7.1.4

Multiply $- 80$ by $60$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 - 80 (- x)}}{2 \cdot 20}$

Step 1.7.1.5

Multiply $- 1$ by $- 80$ .

$y = \frac{120 \pm \sqrt{14400 - 4800 + 80 x}}{2 \cdot 20}$

Step 1.7.1.6

Subtract $4800$ from $14400$ .

$y = \frac{120 \pm \sqrt{9600 + 80 x}}{2 \cdot 20}$

Step 1.7.1.7

Factor $80$ out of $9600 + 80 x$ .

Tap for more steps...

Step 1.7.1.7.1

Factor $80$ out of $9600$ .

$y = \frac{120 \pm \sqrt{80 \cdot 120 + 80 x}}{2 \cdot 20}$

Step 1.7.1.7.2

Factor $80$ out of $80 \cdot 120 + 80 x$ .

$y = \frac{120 \pm \sqrt{80 (120 + x)}}{2 \cdot 20}$

Step 1.7.1.8

Rewrite $80 (120 + x)$ as ${(2^{2})}^{2} \cdot (5 (120 + x))$ .

Tap for more steps...

Step 1.7.1.8.1

Factor $16$ out of $80$ .

$y = \frac{120 \pm \sqrt{16 (5) (120 + x)}}{2 \cdot 20}$

Step 1.7.1.8.2

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

$y = \frac{120 \pm \sqrt{4^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.7.1.8.3

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

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.7.1.8.4

Add parentheses.

$y = \frac{120 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (120 + x))}}{2 \cdot 20}$

Step 1.7.1.9

Pull terms out from under the radical.

$y = \frac{120 \pm 2^{2} \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.7.1.10

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

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{2 \cdot 20}$

Step 1.7.2

Multiply $2$ by $20$ .

$y = \frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$

Step 1.7.3

Simplify $\frac{120 \pm 4 \sqrt{5 (120 + x)}}{40}$ .

$y = \frac{30 \pm \sqrt{5 (120 + x)}}{10}$

Step 1.7.4

Change the $\pm$ to $-$ .

$y = \frac{30 - \sqrt{5 (120 + x)}}{10}$

Step 1.8

The final answer is the combination of both solutions.

$y = \frac{30 + \sqrt{5 (120 + x)}}{10}$

$y = \frac{30 - \sqrt{5 (120 + x)}}{10}$

$y = \frac{30 + \sqrt{5 (120 + x)}}{10}$

$y = \frac{30 - \sqrt{5 (120 + x)}}{10}$

Step 2

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Step 3

Split the fraction $\frac{30 + \sqrt{5 (120 + x)}}{10}$ into two fractions.

$y = \frac{30}{10} + \frac{\sqrt{5 (120 + x)}}{10}$

$y = \frac{30 - \sqrt{5 (120 + x)}}{10}$

Step 4

Divide $30$ by $10$ .

$y = 3 + \frac{\sqrt{5 (120 + x)}}{10}$

$y = \frac{30 - \sqrt{5 (120 + x)}}{10}$

Step 5

Split the fraction $\frac{30 - \sqrt{5 (120 + x)}}{10}$ into two fractions.

$y = 3 + \frac{\sqrt{5 (120 + x)}}{10}$

$y = \frac{30}{10} + \frac{- \sqrt{5 (120 + x)}}{10}$

Step 6

Simplify each term.

Tap for more steps...

Step 6.1

Divide $30$ by $10$ .

$y = 3 + \frac{\sqrt{5 (120 + x)}}{10}$

$y = 3 + \frac{- \sqrt{5 (120 + x)}}{10}$

Step 6.2

Move the negative in front of the fraction.

$y = 3 + \frac{\sqrt{5 (120 + x)}}{10}$

$y = 3 - \frac{\sqrt{5 (120 + x)}}{10}$

$y = 3 + \frac{\sqrt{5 (120 + x)}}{10}$

$y = 3 - \frac{\sqrt{5 (120 + x)}}{10}$

Step 7

Reorder terms.

$y = 3 + \frac{1}{10} \cdot \sqrt{5 (120 + x)}$

$y = 3 - (\frac{1}{10} \cdot \sqrt{5 (120 + x)})$

Step 8

Remove parentheses.

$y = 3 + \frac{1}{10} \cdot \sqrt{5 (120 + x)}$

$y = 3 - \frac{1}{10} \cdot \sqrt{5 (120 + x)}$

Step 9