Algebra Examples

$y = (- 10 + 5 s) (- 9 + 9 s)$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $(- 10 + 5 x) (- 9 + 9 x)$ .

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Step 1.1.1.1

Simplify the expression.

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Step 1.1.1.1.1

Expand $(- 10 + 5 x) (- 9 + 9 x)$ using the FOIL Method.

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Step 1.1.1.1.1.1

Apply the distributive property.

$- 10 (- 9 + 9 x) + 5 x (- 9 + 9 x)$

Step 1.1.1.1.1.2

Apply the distributive property.

$- 10 \cdot - 9 - 10 (9 x) + 5 x (- 9 + 9 x)$

Step 1.1.1.1.1.3

Apply the distributive property.

$- 10 \cdot - 9 - 10 (9 x) + 5 x \cdot - 9 + 5 x (9 x)$

Step 1.1.1.1.2

Simplify and combine like terms.

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Step 1.1.1.1.2.1

Simplify each term.

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Step 1.1.1.1.2.1.1

Multiply $- 10$ by $- 9$ .

$90 - 10 (9 x) + 5 x \cdot - 9 + 5 x (9 x)$

Step 1.1.1.1.2.1.2

Multiply $9$ by $- 10$ .

$90 - 90 x + 5 x \cdot - 9 + 5 x (9 x)$

Step 1.1.1.1.2.1.3

Multiply $- 9$ by $5$ .

$90 - 90 x - 45 x + 5 x (9 x)$

Step 1.1.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$90 - 90 x - 45 x + 5 \cdot 9 x \cdot x$

Step 1.1.1.1.2.1.5

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

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Step 1.1.1.1.2.1.5.1

Move $x$ .

$90 - 90 x - 45 x + 5 \cdot 9 (x \cdot x)$

Step 1.1.1.1.2.1.5.2

Multiply $x$ by $x$ .

$90 - 90 x - 45 x + 5 \cdot 9 x^{2}$

Step 1.1.1.1.2.1.6

Multiply $5$ by $9$ .

$90 - 90 x - 45 x + 45 x^{2}$

Step 1.1.1.1.2.2

Subtract $45 x$ from $- 90 x$ .

$90 - 135 x + 45 x^{2}$

Step 1.1.1.1.3

Move $90$ .

$- 135 x + 45 x^{2} + 90$

Step 1.1.1.1.4

Reorder $- 135 x$ and $45 x^{2}$ .

$45 x^{2} - 135 x + 90$

Step 1.1.1.2

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

$a = 45$

$b = - 135$

$c = 90$

Step 1.1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$d = \frac{- 135}{2 \cdot 45}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Cancel the common factor of $- 135$ and $45$ .

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Step 1.1.1.4.2.1.1

Factor $45$ out of $- 135$ .

$d = \frac{45 \cdot - 3}{2 \cdot 45}$

Step 1.1.1.4.2.1.2

Cancel the common factors.

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Step 1.1.1.4.2.1.2.1

Factor $45$ out of $2 \cdot 45$ .

$d = \frac{45 \cdot - 3}{45 \cdot 2}$

Step 1.1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{45 \cdot - 3}{45 \cdot 2}$

Step 1.1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 3}{2}$

Step 1.1.1.4.2.2

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

Step 1.1.1.5

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

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Step 1.1.1.5.1

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

$e = 90 - \frac{{(- 135)}^{2}}{4 \cdot 45}$

Step 1.1.1.5.2

Simplify the right side.

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Step 1.1.1.5.2.1

Simplify each term.

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Step 1.1.1.5.2.1.1

Raise $- 135$ to the power of $2$ .

$e = 90 - \frac{18225}{4 \cdot 45}$

Step 1.1.1.5.2.1.2

Multiply $4$ by $45$ .

$e = 90 - \frac{18225}{180}$

Step 1.1.1.5.2.1.3

Cancel the common factor of $18225$ and $180$ .

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Step 1.1.1.5.2.1.3.1

Factor $45$ out of $18225$ .

$e = 90 - \frac{45 (405)}{180}$

Step 1.1.1.5.2.1.3.2

Cancel the common factors.

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Step 1.1.1.5.2.1.3.2.1

Factor $45$ out of $180$ .

$e = 90 - \frac{45 \cdot 405}{45 \cdot 4}$

Step 1.1.1.5.2.1.3.2.2

Cancel the common factor.

$e = 90 - \frac{45 \cdot 405}{45 \cdot 4}$

Step 1.1.1.5.2.1.3.2.3

Rewrite the expression.

$e = 90 - \frac{405}{4}$

Step 1.1.1.5.2.2

To write $90$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 90 \cdot \frac{4}{4} - \frac{405}{4}$

Step 1.1.1.5.2.3

Combine $90$ and $\frac{4}{4}$ .

$e = \frac{90 \cdot 4}{4} - \frac{405}{4}$

Step 1.1.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{90 \cdot 4 - 405}{4}$

Step 1.1.1.5.2.5

Simplify the numerator.

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Step 1.1.1.5.2.5.1

Multiply $90$ by $4$ .

$e = \frac{360 - 405}{4}$

Step 1.1.1.5.2.5.2

Subtract $405$ from $360$ .

$e = \frac{- 45}{4}$

Step 1.1.1.5.2.6

Move the negative in front of the fraction.

$e = - \frac{45}{4}$

Step 1.1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $45 {(x - \frac{3}{2})}^{2} - \frac{45}{4}$ .

$45 {(x - \frac{3}{2})}^{2} - \frac{45}{4}$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 45 {(x - \frac{3}{2})}^{2} - \frac{45}{4}$

Step 1.2

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

$a = 45$

$h = \frac{3}{2}$

$k = - \frac{45}{4}$

Step 1.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(\frac{3}{2}, - \frac{45}{4})$

Step 1.5

Find $p$ , the distance from the vertex to the focus.

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Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 45}$

Step 1.5.3

Multiply $4$ by $45$ .

$\frac{1}{180}$

Step 1.6

Find the focus.

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Step 1.6.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(\frac{3}{2}, - \frac{506}{45})$

Step 1.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = \frac{3}{2}$

Step 1.8

Find the directrix.

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Step 1.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.8.2

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = - \frac{1013}{90}$

Step 1.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{45}{4})$

Focus: $(\frac{3}{2}, - \frac{506}{45})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1013}{90}$

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{45}{4})$

Focus: $(\frac{3}{2}, - \frac{506}{45})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1013}{90}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = (- 10 + 5 (1)) (- 9 + 9 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $5$ by $1$ .

$f (1) = (- 10 + 5) (- 9 + 9 (1))$

Step 2.2.2

Add $- 10$ and $5$ .

$f (1) = - 5 (- 9 + 9 (1))$

Step 2.2.3

Multiply $9$ by $1$ .

$f (1) = - 5 (- 9 + 9)$

Step 2.2.4

Add $- 9$ and $9$ .

$f (1) = - 5 \cdot 0$

Step 2.2.5

Multiply $- 5$ by $0$ .

$f (1) = 0$

Step 2.2.6

The final answer is $0$ .

$0$

Step 2.3

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Step 2.4

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & 0 \\ \frac{3}{2} & - \frac{45}{4} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(\frac{3}{2}, - \frac{45}{4})$

Focus: $(\frac{3}{2}, - \frac{506}{45})$

Axis of Symmetry: $x = \frac{3}{2}$

Directrix: $y = - \frac{1013}{90}$

$\begin{array}{c} x & y \\ 1 & 0 \\ \frac{3}{2} & - \frac{45}{4} \end{array}$

Step 4