Precalculus Examples

$y = 3 {(x - 2)}^{2} - {(x - 5)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

$y = 3 {(x - 2)}^{2} - ((x - 5) (x - 5))$

Step 1.1.1.2

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

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Step 1.1.1.2.1

Apply the distributive property.

$y = 3 {(x - 2)}^{2} - (x (x - 5) - 5 (x - 5))$

Step 1.1.1.2.2

Apply the distributive property.

$y = 3 {(x - 2)}^{2} - (x \cdot x + x \cdot - 5 - 5 (x - 5))$

Step 1.1.1.2.3

Apply the distributive property.

$y = 3 {(x - 2)}^{2} - (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$y = 3 {(x - 2)}^{2} - (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Step 1.1.1.3.1.2

Move $- 5$ to the left of $x$ .

$y = 3 {(x - 2)}^{2} - (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)$

Step 1.1.1.3.1.3

Multiply $- 5$ by $- 5$ .

$y = 3 {(x - 2)}^{2} - (x^{2} - 5 x - 5 x + 25)$

Step 1.1.1.3.2

Subtract $5 x$ from $- 5 x$ .

$y = 3 {(x - 2)}^{2} - (x^{2} - 10 x + 25)$

Step 1.1.1.4

Apply the distributive property.

$y = 3 {(x - 2)}^{2} - x^{2} - (- 10 x) - 1 \cdot 25$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Multiply $- 10$ by $- 1$ .

$y = 3 {(x - 2)}^{2} - x^{2} + 10 x - 1 \cdot 25$

Step 1.1.1.5.2

Multiply $- 1$ by $25$ .

$y = 3 {(x - 2)}^{2} - x^{2} + 10 x - 25$

Step 1.1.2

Reorder terms.

$y = - x^{2} + 3 {(x - 2)}^{2} + 10 x - 25$

Step 1.2

Complete the square for $- x^{2} + 3 {(x - 2)}^{2} + 10 x - 25$ .

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

$- x^{2} + 3 ((x - 2) (x - 2)) + 10 x - 25$

Step 1.2.1.1.2

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

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Step 1.2.1.1.2.1

Apply the distributive property.

$- x^{2} + 3 (x (x - 2) - 2 (x - 2)) + 10 x - 25$

Step 1.2.1.1.2.2

Apply the distributive property.

$- x^{2} + 3 (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 10 x - 25$

Step 1.2.1.1.2.3

Apply the distributive property.

$- x^{2} + 3 (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 10 x - 25$

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 $x$ by $x$ .

$- x^{2} + 3 (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + 10 x - 25$

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $x$ .

$- x^{2} + 3 (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + 10 x - 25$

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$- x^{2} + 3 (x^{2} - 2 x - 2 x + 4) + 10 x - 25$

Step 1.2.1.1.3.2

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

$- x^{2} + 3 (x^{2} - 4 x + 4) + 10 x - 25$

Step 1.2.1.1.4

Apply the distributive property.

$- x^{2} + 3 x^{2} + 3 (- 4 x) + 3 \cdot 4 + 10 x - 25$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 4$ by $3$ .

$- x^{2} + 3 x^{2} - 12 x + 3 \cdot 4 + 10 x - 25$

Step 1.2.1.1.5.2

Multiply $3$ by $4$ .

$- x^{2} + 3 x^{2} - 12 x + 12 + 10 x - 25$

Step 1.2.1.2

Add $- x^{2}$ and $3 x^{2}$ .

$2 x^{2} - 12 x + 12 + 10 x - 25$

Step 1.2.1.3

Add $- 12 x$ and $10 x$ .

$2 x^{2} - 2 x + 12 - 25$

Step 1.2.1.4

Subtract $25$ from $12$ .

$2 x^{2} - 2 x - 13$

Step 1.2.2

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

$a = 2$

$b = - 2$

$c = - 13$

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

Factor $2$ out of $- 2$ .

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

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 $2 \cdot 2$ .

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

Step 1.2.4.2.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.2

Move the negative in front of the fraction.

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

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

$e = - 13 - \frac{{(- 2)}^{2}}{4 \cdot 2}$

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 ${(- 2)}^{2}$ and $2$ .

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Step 1.2.5.2.1.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$e = - 13 - \frac{{(- 1 (2))}^{2}}{4 \cdot 2}$

Step 1.2.5.2.1.1.2

Apply the product rule to $- 1 (2)$ .

$e = - 13 - \frac{{(- 1)}^{2} \cdot 2^{2}}{4 \cdot 2}$

Step 1.2.5.2.1.1.3

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

$e = - 13 - \frac{1 \cdot 2^{2}}{4 \cdot 2}$

Step 1.2.5.2.1.1.4

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

$e = - 13 - \frac{2^{2}}{4 \cdot 2}$

Step 1.2.5.2.1.1.5

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

$e = - 13 - \frac{2 \cdot 2}{4 \cdot 2}$

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

$e = - 13 - \frac{2 \cdot 2}{2 (2 \cdot 2)}$

Step 1.2.5.2.1.1.6.2

Cancel the common factor.

$e = - 13 - \frac{2 \cdot 2}{2 (2 \cdot 2)}$

Step 1.2.5.2.1.1.6.3

Rewrite the expression.

$e = - 13 - \frac{2}{2 \cdot 2}$

Step 1.2.5.2.1.2

Cancel the common factor of $2$ .

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Step 1.2.5.2.1.2.1

Cancel the common factor.

$e = - 13 - \frac{2}{2 \cdot 2}$

Step 1.2.5.2.1.2.2

Rewrite the expression.

$e = - 13 - \frac{1}{2}$

Step 1.2.5.2.2

To write $- 13$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 13 \cdot \frac{2}{2} - \frac{1}{2}$

Step 1.2.5.2.3

Combine $- 13$ and $\frac{2}{2}$ .

$e = \frac{- 13 \cdot 2}{2} - \frac{1}{2}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 13 \cdot 2 - 1}{2}$

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $- 13$ by $2$ .

$e = \frac{- 26 - 1}{2}$

Step 1.2.5.2.5.2

Subtract $1$ from $- 26$ .

$e = \frac{- 27}{2}$

Step 1.2.5.2.6

Move the negative in front of the fraction.

$e = - \frac{27}{2}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - \frac{1}{2})}^{2} - \frac{27}{2}$ .

$2 {(x - \frac{1}{2})}^{2} - \frac{27}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = 2 {(x - \frac{1}{2})}^{2} - \frac{27}{2}$

Step 2

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

$a = 2$

$h = \frac{1}{2}$

$k = - \frac{27}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, - \frac{27}{2})$

Step 4