Algebra Examples

$(10, - 5)$ , $(5, 0)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(10, - 5)$ as the vertex $(h, k)$ and $(5, 0)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

$0 = a {(5 - (10))}^{2} - 5$

Step 2

Using $0 = a {(5 - (10))}^{2} - 5$ to solve for $a$ , $a = \frac{1}{5}$ .

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Step 2.1

Rewrite the equation as $a {(5 - (10))}^{2} - 5 = 0$ .

$a {(5 - (10))}^{2} - 5 = 0$

Step 2.2

Simplify each term.

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Step 2.2.1

Multiply $- 1$ by $10$ .

$a {(5 - 10)}^{2} - 5 = 0$

Step 2.2.2

Subtract $10$ from $5$ .

$a {(- 5)}^{2} - 5 = 0$

Step 2.2.3

Raise $- 5$ to the power of $2$ .

$a \cdot 25 - 5 = 0$

Step 2.2.4

Move $25$ to the left of $a$ .

$25 a - 5 = 0$

Step 2.3

Add $5$ to both sides of the equation.

$25 a = 5$

Step 2.4

Divide each term in $25 a = 5$ by $25$ and simplify.

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Step 2.4.1

Divide each term in $25 a = 5$ by $25$ .

$\frac{25 a}{25} = \frac{5}{25}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $25$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{25 a}{25} = \frac{5}{25}$

Step 2.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{5}{25}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $5$ and $25$ .

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Step 2.4.3.1.1

Factor $5$ out of $5$ .

$a = \frac{5 (1)}{25}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $5$ out of $25$ .

$a = \frac{5 \cdot 1}{5 \cdot 5}$

Step 2.4.3.1.2.2

Cancel the common factor.

$a = \frac{5 \cdot 1}{5 \cdot 5}$

Step 2.4.3.1.2.3

Rewrite the expression.

$a = \frac{1}{5}$

Step 3

Using $y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex $(10, - 5)$ and $a = \frac{1}{5}$ is $y = (\frac{1}{5}) {(x - (10))}^{2} - 5$ .

$y = (\frac{1}{5}) {(x - (10))}^{2} - 5$

Step 4

Solve $y = (\frac{1}{5}) {(x - (10))}^{2} - 5$ for $y$ .

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Step 4.1

Remove parentheses.

$y = (\frac{1}{5}) {(x - (10))}^{2} - 5$

Step 4.2

Multiply $\frac{1}{5}$ by ${(x - (10))}^{2}$ .

$y = \frac{1}{5} \cdot {(x - (10))}^{2} - 5$

Step 4.3

Remove parentheses.

$y = (\frac{1}{5}) {(x - (10))}^{2} - 5$

Step 4.4

Simplify $(\frac{1}{5}) {(x - (10))}^{2} - 5$ .

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Step 4.4.1

Simplify each term.

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Step 4.4.1.1

Multiply $- 1$ by $10$ .

$y = \frac{1}{5} {(x - 10)}^{2} - 5$

Step 4.4.1.2

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$y = \frac{1}{5} ((x - 10) (x - 10)) - 5$

Step 4.4.1.3

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

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Step 4.4.1.3.1

Apply the distributive property.

$y = \frac{1}{5} (x (x - 10) - 10 (x - 10)) - 5$

Step 4.4.1.3.2

Apply the distributive property.

$y = \frac{1}{5} (x \cdot x + x \cdot - 10 - 10 (x - 10)) - 5$

Step 4.4.1.3.3

Apply the distributive property.

$y = \frac{1}{5} (x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10) - 5$

Step 4.4.1.4

Simplify and combine like terms.

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Step 4.4.1.4.1

Simplify each term.

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Step 4.4.1.4.1.1

Multiply $x$ by $x$ .

$y = \frac{1}{5} (x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10) - 5$

Step 4.4.1.4.1.2

Move $- 10$ to the left of $x$ .

$y = \frac{1}{5} (x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10) - 5$

Step 4.4.1.4.1.3

Multiply $- 10$ by $- 10$ .

$y = \frac{1}{5} (x^{2} - 10 x - 10 x + 100) - 5$

Step 4.4.1.4.2

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

$y = \frac{1}{5} (x^{2} - 20 x + 100) - 5$

Step 4.4.1.5

Apply the distributive property.

$y = \frac{1}{5} x^{2} + \frac{1}{5} (- 20 x) + \frac{1}{5} \cdot 100 - 5$

Step 4.4.1.6

Simplify.

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Step 4.4.1.6.1

Combine $\frac{1}{5}$ and $x^{2}$ .

$y = \frac{x^{2}}{5} + \frac{1}{5} (- 20 x) + \frac{1}{5} \cdot 100 - 5$

Step 4.4.1.6.2

Cancel the common factor of $5$ .

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Step 4.4.1.6.2.1

Factor $5$ out of $- 20 x$ .

$y = \frac{x^{2}}{5} + \frac{1}{5} (5 (- 4 x)) + \frac{1}{5} \cdot 100 - 5$

Step 4.4.1.6.2.2

Cancel the common factor.

$y = \frac{x^{2}}{5} + \frac{1}{5} (5 (- 4 x)) + \frac{1}{5} \cdot 100 - 5$

Step 4.4.1.6.2.3

Rewrite the expression.

$y = \frac{x^{2}}{5} - 4 x + \frac{1}{5} \cdot 100 - 5$

Step 4.4.1.6.3

Cancel the common factor of $5$ .

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Step 4.4.1.6.3.1

Factor $5$ out of $100$ .

$y = \frac{x^{2}}{5} - 4 x + \frac{1}{5} \cdot (5 (20)) - 5$

Step 4.4.1.6.3.2

Cancel the common factor.

$y = \frac{x^{2}}{5} - 4 x + \frac{1}{5} \cdot (5 \cdot 20) - 5$

Step 4.4.1.6.3.3

Rewrite the expression.

$y = \frac{x^{2}}{5} - 4 x + 20 - 5$

Step 4.4.2

Subtract $5$ from $20$ .

$y = \frac{x^{2}}{5} - 4 x + 15$

Step 5

The standard form and vertex form are as follows.

Standard Form: $y = \frac{1}{5} x^{2} - 4 x + 15$

Vertex Form: $y = (\frac{1}{5}) {(x - (10))}^{2} - 5$

Step 6

Simplify the standard form.

Standard Form: $y = \frac{1}{5} x^{2} - 4 x + 15$

Vertex Form: $y = \frac{1}{5} {(x - 10)}^{2} - 5$

Step 7