Precalculus Examples

$y - 3 = {(2 - x)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Add $3$ to both sides of the equation.

$y = {(2 - x)}^{2} + 3$

Step 1.2

Complete the square for ${(2 - x)}^{2} + 3$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(2 - x)}^{2}$ as $(2 - x) (2 - x)$ .

$(2 - x) (2 - x) + 3$

Step 1.2.1.1.2

Expand $(2 - x) (2 - x)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$2 (2 - x) - x (2 - x) + 3$

Step 1.2.1.1.2.2

Apply the distributive property.

$2 \cdot 2 + 2 (- x) - x (2 - x) + 3$

Step 1.2.1.1.2.3

Apply the distributive property.

$2 \cdot 2 + 2 (- x) - x \cdot 2 - x (- x) + 3$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $2$ by $2$ .

$4 + 2 (- x) - x \cdot 2 - x (- x) + 3$

Step 1.2.1.1.3.1.2

Multiply $- 1$ by $2$ .

$4 - 2 x - x \cdot 2 - x (- x) + 3$

Step 1.2.1.1.3.1.3

Multiply $2$ by $- 1$ .

$4 - 2 x - 2 x - x (- x) + 3$

Step 1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 - 2 x - 2 x - 1 \cdot - 1 x \cdot x + 3$

Step 1.2.1.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1.1.3.1.5.1

Move $x$ .

$4 - 2 x - 2 x - 1 \cdot - 1 (x \cdot x) + 3$

Step 1.2.1.1.3.1.5.2

Multiply $x$ by $x$ .

$4 - 2 x - 2 x - 1 \cdot - 1 x^{2} + 3$

Step 1.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 - 2 x - 2 x + 1 x^{2} + 3$

Step 1.2.1.1.3.1.7

Multiply $x^{2}$ by $1$ .

$4 - 2 x - 2 x + x^{2} + 3$

Step 1.2.1.1.3.2

Subtract $2 x$ from $- 2 x$ .

$4 - 4 x + x^{2} + 3$

Step 1.2.1.2

Add $4$ and $3$ .

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

Step 1.2.1.3

Reorder $- 4 x$ and $x^{2}$ .

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

Step 1.2.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 4$

$c = 7$

Step 1.2.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot 1}$

Step 1.2.4.2

Cancel the common factor of $- 4$ and $2$ .

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Step 1.2.4.2.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 1.2.4.2.2

Cancel the common factors.

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Step 1.2.4.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 2}{2 (1)}$

Step 1.2.4.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 1.2.4.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 1.2.4.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 1.2.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 7 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.2.5.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 7 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 1.2.5.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 7 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 1.2.5.2.1.1.3

Raise $- 1$ to the power of $2$ .

$e = 7 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 1.2.5.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 7 - \frac{4^{2}}{4 \cdot 1}$

Step 1.2.5.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.2.5.2.1.1.6

Cancel the common factors.

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Step 1.2.5.2.1.1.6.1

Factor $4$ out of $4 \cdot 1$ .

$e = 7 - \frac{4 \cdot 4}{4 (1)}$

Step 1.2.5.2.1.1.6.2

Cancel the common factor.

$e = 7 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.2.5.2.1.1.6.3

Rewrite the expression.

$e = 7 - \frac{4}{1}$

Step 1.2.5.2.1.1.6.4

Divide $4$ by $1$ .

$e = 7 - 1 \cdot 4$

Step 1.2.5.2.1.2

Multiply $- 1$ by $4$ .

$e = 7 - 4$

Step 1.2.5.2.2

Subtract $4$ from $7$ .

$e = 3$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 2)}^{2} + 3$ .

${(x - 2)}^{2} + 3$

Step 1.3

Set $y$ equal to the new right side.

$y = {(x - 2)}^{2} + 3$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = 1$

$h = 2$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(2, 3)$

Step 4