Precalculus Examples

$y + 6 = {(2 x - 5)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Subtract $6$ from both sides of the equation.

$y = {(2 x - 5)}^{2} - 6$

Step 1.2

Complete the square for ${(2 x - 5)}^{2} - 6$ .

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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 ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$(2 x - 5) (2 x - 5) - 6$

Step 1.2.1.1.2

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$2 x (2 x - 5) - 5 (2 x - 5) - 6$

Step 1.2.1.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5) - 6$

Step 1.2.1.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 6$

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

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.2

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

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Step 1.2.1.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.4

Multiply $- 5$ by $2$ .

$4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.5

Multiply $2$ by $- 5$ .

$4 x^{2} - 10 x - 10 x - 5 \cdot - 5 - 6$

Step 1.2.1.1.3.1.6

Multiply $- 5$ by $- 5$ .

$4 x^{2} - 10 x - 10 x + 25 - 6$

Step 1.2.1.1.3.2

Subtract $10 x$ from $- 10 x$ .

$4 x^{2} - 20 x + 25 - 6$

Step 1.2.1.2

Subtract $6$ from $25$ .

$4 x^{2} - 20 x + 19$

Step 1.2.2

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

$a = 4$

$b = - 20$

$c = 19$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 20}{2 \cdot 4}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $- 20$ and $2$ .

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Step 1.2.4.2.1.1

Factor $2$ out of $- 20$ .

$d = \frac{2 \cdot - 10}{2 \cdot 4}$

Step 1.2.4.2.1.2

Cancel the common factors.

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Step 1.2.4.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 10}{2 (4)}$

Step 1.2.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 10}{2 \cdot 4}$

Step 1.2.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 10}{4}$

Step 1.2.4.2.2

Cancel the common factor of $- 10$ and $4$ .

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Step 1.2.4.2.2.1

Factor $2$ out of $- 10$ .

$d = \frac{2 (- 5)}{4}$

Step 1.2.4.2.2.2

Cancel the common factors.

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Step 1.2.4.2.2.2.1

Factor $2$ out of $4$ .

$d = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 1.2.4.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 2}$

Step 1.2.4.2.2.2.3

Rewrite the expression.

$d = \frac{- 5}{2}$

Step 1.2.4.2.3

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 19 - \frac{{(- 20)}^{2}}{4 \cdot 4}$

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

Raise $- 20$ to the power of $2$ .

$e = 19 - \frac{400}{4 \cdot 4}$

Step 1.2.5.2.1.2

Multiply $4$ by $4$ .

$e = 19 - \frac{400}{16}$

Step 1.2.5.2.1.3

Divide $400$ by $16$ .

$e = 19 - 1 \cdot 25$

Step 1.2.5.2.1.4

Multiply $- 1$ by $25$ .

$e = 19 - 25$

Step 1.2.5.2.2

Subtract $25$ from $19$ .

$e = - 6$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x - \frac{5}{2})}^{2} - 6$ .

$4 {(x - \frac{5}{2})}^{2} - 6$

Step 1.3

Set $y$ equal to the new right side.

$y = 4 {(x - \frac{5}{2})}^{2} - 6$

Step 2

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

$a = 4$

$h = \frac{5}{2}$

$k = - 6$

Step 3

Find the vertex $(h, k)$ .

$(\frac{5}{2}, - 6)$

Step 4