Trigonometry Examples

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

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

$x$ .

$0 = 2 {(x + 3)}^{2} + 10$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$0 = 2 {(x + 3)}^{2} + 10$

Step 1.2.2

Simplify

$2 {(x + 3)}^{2} + 10$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Rewrite

${(x + 3)}^{2}$ as

$(x + 3) (x + 3)$ .

$0 = 2 ((x + 3) (x + 3)) + 10$

Step 1.2.2.1.2

Expand

$(x + 3) (x + 3)$ using the FOIL Method.

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Step 1.2.2.1.2.1

Apply the distributive property.

$0 = 2 (x (x + 3) + 3 (x + 3)) + 10$

Step 1.2.2.1.2.2

Apply the distributive property.

$0 = 2 (x \cdot x + x \cdot 3 + 3 (x + 3)) + 10$

Step 1.2.2.1.2.3

Apply the distributive property.

$0 = 2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 10$

$0 = 2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) + 10$

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

$x$ .

$0 = 2 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) + 10$

Step 1.2.2.1.3.1.2

Move

$3$ to the left of

$x$ .

$0 = 2 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) + 10$

Step 1.2.2.1.3.1.3

Multiply

$3$ by

$3$ .

$0 = 2 (x^{2} + 3 x + 3 x + 9) + 10$

$0 = 2 (x^{2} + 3 x + 3 x + 9) + 10$

Step 1.2.2.1.3.2

Add

$3 x$ and

$3 x$ .

$0 = 2 (x^{2} + 6 x + 9) + 10$

$0 = 2 (x^{2} + 6 x + 9) + 10$

Step 1.2.2.1.4

Apply the distributive property.

$0 = 2 x^{2} + 2 (6 x) + 2 \cdot 9 + 10$

Step 1.2.2.1.5

Simplify.

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Step 1.2.2.1.5.1

Multiply

$6$ by

$2$ .

$0 = 2 x^{2} + 12 x + 2 \cdot 9 + 10$

Step 1.2.2.1.5.2

Multiply

$2$ by

$9$ .

$0 = 2 x^{2} + 12 x + 18 + 10$

$0 = 2 x^{2} + 12 x + 18 + 10$

$0 = 2 x^{2} + 12 x + 18 + 10$

Step 1.2.2.2

Add

$18$ and

$10$ .

$0 = 2 x^{2} + 12 x + 28$

$0 = 2 x^{2} + 12 x + 28$

Step 1.2.3

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

No solution

No solution

Step 1.3

To find the x-intercept(s), substitute in

$0$ for

$y$ and solve for

$x$ .

x-intercept(s):

$None$

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

$y$ .

$y = 2 {((0) + 3)}^{2} + 10$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 {(0 + 3)}^{2} + 10$

Step 2.2.2

Remove parentheses.

$y = 2 {((0) + 3)}^{2} + 10$

Step 2.2.3

Simplify

$2 {((0) + 3)}^{2} + 10$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Add

$0$ and

$3$ .

$y = 2 \cdot 3^{2} + 10$

Step 2.2.3.1.2

Raise

$3$ to the power of

$2$ .

$y = 2 \cdot 9 + 10$

Step 2.2.3.1.3

Multiply

$2$ by

$9$ .

$y = 18 + 10$

$y = 18 + 10$

Step 2.2.3.2

Add

$18$ and

$10$ .

$y = 28$

$y = 28$

$y = 28$

Step 2.3

y-intercept(s) in point form.

y-intercept(s):

$(0, 28)$

y-intercept(s):

$(0, 28)$

Step 3

List the intersections.

x-intercept(s):

$None$

y-intercept(s):

$(0, 28)$

Step 4

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$