Finite Math Examples

$y = 4 {(x - 5)}^{2} + 1$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 {(x - 5)}^{2} + 1$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 4 {(x - 5)}^{2} + 1$

Step 1.2.2

Simplify $4 {(x - 5)}^{2} + 1$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

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

$0 = 4 ((x - 5) (x - 5)) + 1$

Step 1.2.2.1.2

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

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Step 1.2.2.1.2.1

Apply the distributive property.

$0 = 4 (x (x - 5) - 5 (x - 5)) + 1$

Step 1.2.2.1.2.2

Apply the distributive property.

$0 = 4 (x \cdot x + x \cdot - 5 - 5 (x - 5)) + 1$

Step 1.2.2.1.2.3

Apply the distributive property.

$0 = 4 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + 1$

Step 1.2.2.1.3

Simplify and combine like terms.

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Step 1.2.2.1.3.1

Simplify each term.

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Step 1.2.2.1.3.1.1

Multiply $x$ by $x$ .

$0 = 4 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + 1$

Step 1.2.2.1.3.1.2

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

$0 = 4 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + 1$

Step 1.2.2.1.3.1.3

Multiply $- 5$ by $- 5$ .

$0 = 4 (x^{2} - 5 x - 5 x + 25) + 1$

Step 1.2.2.1.3.2

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

$0 = 4 (x^{2} - 10 x + 25) + 1$

Step 1.2.2.1.4

Apply the distributive property.

$0 = 4 x^{2} + 4 (- 10 x) + 4 \cdot 25 + 1$

Step 1.2.2.1.5

Simplify.

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Step 1.2.2.1.5.1

Multiply $- 10$ by $4$ .

$0 = 4 x^{2} - 40 x + 4 \cdot 25 + 1$

Step 1.2.2.1.5.2

Multiply $4$ by $25$ .

$0 = 4 x^{2} - 40 x + 100 + 1$

Step 1.2.2.2

Add $100$ and $1$ .

$0 = 4 x^{2} - 40 x + 101$

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4 {((0) - 5)}^{2} + 1$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 {(0 - 5)}^{2} + 1$

Step 2.2.2

Remove parentheses.

$y = 4 {((0) - 5)}^{2} + 1$

Step 2.2.3

Simplify $4 {((0) - 5)}^{2} + 1$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Subtract $5$ from $0$ .

$y = 4 {(- 5)}^{2} + 1$

Step 2.2.3.1.2

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

$y = 4 \cdot 25 + 1$

Step 2.2.3.1.3

Multiply $4$ by $25$ .

$y = 100 + 1$

Step 2.2.3.2

Add $100$ and $1$ .

$y = 101$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 101)$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, 101)$

Step 4