Precalculus Examples

$y^{2} - 4 y + 4 = x - 7$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms to the left side of the equation and simplify.

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Step 1.1.1

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

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Step 1.1.1.1

Subtract $x$ from both sides of the equation.

$y^{2} - 4 y + 4 - x = - 7$

Step 1.1.1.2

Add $7$ to both sides of the equation.

$y^{2} - 4 y + 4 - x + 7 = 0$

Step 1.1.2

Add $4$ and $7$ .

$y^{2} - 4 y - x + 11 = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 1$ , $b = - 4$ , and $c = - x + 11$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- x + 11)$ .

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Step 1.4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- x + 11)}}{2 \cdot 1}$

Step 1.4.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- x + 11)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.4.1.2

Multiply $- x + 11$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x + 11)}}{2 \cdot 1}$

Step 1.4.1.3

Add $- 4$ and $11$ .

$y = \frac{4 \pm \sqrt{- 4 (- x + 7)}}{2 \cdot 1}$

Step 1.4.1.4

Rewrite $- 4 (- x + 7)$ as $2^{2} \cdot (- 1 (- x + 7))$ .

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Step 1.4.1.4.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- x + 7)}}{2 \cdot 1}$

Step 1.4.1.4.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.4.1.4.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.4.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2 \cdot 1}$

Step 1.4.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$

Step 1.4.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- x + 7)}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- x + 11)$ .

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Step 1.5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- x + 11)}}{2 \cdot 1}$

Step 1.5.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- x + 11)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.5.1.2

Multiply $- x + 11$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x + 11)}}{2 \cdot 1}$

Step 1.5.1.3

Add $- 4$ and $11$ .

$y = \frac{4 \pm \sqrt{- 4 (- x + 7)}}{2 \cdot 1}$

Step 1.5.1.4

Rewrite $- 4 (- x + 7)$ as $2^{2} \cdot (- 1 (- x + 7))$ .

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Step 1.5.1.4.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- x + 7)}}{2 \cdot 1}$

Step 1.5.1.4.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.5.1.4.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.5.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2 \cdot 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$

Step 1.5.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- x + 7)}$

Step 1.5.4

Change the $\pm$ to $+$ .

$y = 2 + \sqrt{- 1 (- x + 7)}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- x + 11)$ .

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Step 1.6.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- x + 11)}}{2 \cdot 1}$

Step 1.6.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- x + 11)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- x + 11))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- x + 11))}}{2 \cdot 1}$

Step 1.6.1.2

Multiply $- x + 11$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x + 11)}}{2 \cdot 1}$

Step 1.6.1.3

Add $- 4$ and $11$ .

$y = \frac{4 \pm \sqrt{- 4 (- x + 7)}}{2 \cdot 1}$

Step 1.6.1.4

Rewrite $- 4 (- x + 7)$ as $2^{2} \cdot (- 1 (- x + 7))$ .

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Step 1.6.1.4.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- x + 7)}}{2 \cdot 1}$

Step 1.6.1.4.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.6.1.4.3

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- x + 7))}}{2 \cdot 1}$

Step 1.6.1.5

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2 \cdot 1}$

Step 1.6.2

Multiply $2$ by $1$ .

$y = \frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$

Step 1.6.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- x + 7)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- x + 7)}$

Step 1.6.4

Change the $\pm$ to $-$ .

$y = 2 - \sqrt{- 1 (- x + 7)}$

Step 1.7

The final answer is the combination of both solutions.

$y = 2 + \sqrt{- 1 (- x + 7)}$

$y = 2 - \sqrt{- 1 (- x + 7)}$

$y = 2 + \sqrt{- 1 (- x + 7)}$

$y = 2 - \sqrt{- 1 (- x + 7)}$

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

Rewrite $- 1 (- x + 7)$ as $- (- x + 7)$ .

$y = 2 + \sqrt{- (- x + 7)}$

$y = 2 - \sqrt{- (- x + 7)}$

Step 4