Trigonometry Examples

$\sqrt{9 - x} = x + 3$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{9 - x}}^{2} = {(x + 3)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 - x}$ as ${(9 - x)}^{\frac{1}{2}}$ .

${({(9 - x)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(9 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(9 - x)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(9 - x)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 - x)}^{1} = {(x + 3)}^{2}$

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

$9 - x = (x + 3) (x + 3)$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Apply the distributive property.

$9 - x = x (x + 3) + 3 (x + 3)$

Step 2.3.1.2.2

Apply the distributive property.

$9 - x = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 2.3.1.2.3

Apply the distributive property.

$9 - x = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$9 - x = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.2

Move $3$ to the left of $x$ .

$9 - x = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.3

Multiply $3$ by $3$ .

$9 - x = x^{2} + 3 x + 3 x + 9$

Step 2.3.1.3.2

Add $3 x$ and $3 x$ .

$9 - x = x^{2} + 6 x + 9$

Step 3

Solve for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 6 x + 9 = 9 - x$

Step 3.2

Move all terms containing $x$ to the left side of the equation.

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Step 3.2.1

Add $x$ to both sides of the equation.

$x^{2} + 6 x + 9 + x = 9$

Step 3.2.2

Add $6 x$ and $x$ .

$x^{2} + 7 x + 9 = 9$

Step 3.3

Subtract $9$ from both sides of the equation.

$x^{2} + 7 x + 9 - 9 = 0$

Step 3.4

Combine the opposite terms in $x^{2} + 7 x + 9 - 9$ .

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Step 3.4.1

Subtract $9$ from $9$ .

$x^{2} + 7 x + 0 = 0$

Step 3.4.2

Add $x^{2} + 7 x$ and $0$ .

$x^{2} + 7 x = 0$

Step 3.5

Factor $x$ out of $x^{2} + 7 x$ .

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Step 3.5.1

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

$x \cdot x + 7 x = 0$

Step 3.5.2

Factor $x$ out of $7 x$ .

$x \cdot x + x \cdot 7 = 0$

Step 3.5.3

Factor $x$ out of $x \cdot x + x \cdot 7$ .

$x (x + 7) = 0$

Step 3.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 7 = 0$

Step 3.7

Set $x$ equal to $0$ .

$x = 0$

Step 3.8

Set $x + 7$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 3.8.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 3.9

The final solution is all the values that make $x (x + 7) = 0$ true.

$x = 0, - 7$

Step 4

Exclude the solutions that do not make $\sqrt{9 - x} = x + 3$ true.

$x = 0$