Trigonometry Examples

$x^{2} + y^{2} - 4 x = 0$

Step 1

Solve for $y$ .

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$y^{2} - 4 x = - x^{2}$

Step 1.1.2

Add $4 x$ to both sides of the equation.

$y^{2} = - x^{2} + 4 x$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{- x^{2} + 4 x}$

Step 1.3

Factor $x$ out of $- x^{2} + 4 x$ .

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Step 1.3.1

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

$y = \pm \sqrt{x (- x) + 4 x}$

Step 1.3.2

Factor $x$ out of $4 x$ .

$y = \pm \sqrt{x (- x) + x \cdot 4}$

Step 1.3.3

Factor $x$ out of $x (- x) + x \cdot 4$ .

$y = \pm \sqrt{x (- x + 4)}$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{x (- x + 4)}$

Step 1.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{x (- x + 4)}$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{x (- x + 4)}$

$y = - \sqrt{x (- x + 4)}$

$y = \sqrt{x (- x + 4)}$

$y = - \sqrt{x (- x + 4)}$

$y = \sqrt{x (- x + 4)}$

$y = - \sqrt{x (- x + 4)}$

Step 2

Find the domain to determine if the expression is continuous.

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Step 2.1

Set the radicand in $\sqrt{x (- x + 4)}$ greater than or equal to $0$ to find where the expression is defined.

$x (- x + 4) \geq 0$

Step 2.2

Solve for $x$ .

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Step 2.2.1

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

$x = 0$

$- x + 4 = 0$

Step 2.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.3

Set $- x + 4$ equal to $0$ and solve for $x$ .

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Step 2.2.3.1

Set $- x + 4$ equal to $0$ .

$- x + 4 = 0$

Step 2.2.3.2

Solve $- x + 4 = 0$ for $x$ .

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Step 2.2.3.2.1

Subtract $4$ from both sides of the equation.

$- x = - 4$

Step 2.2.3.2.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 2.2.3.2.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 2.2.3.2.2.2

Simplify the left side.

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Step 2.2.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 2.2.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 2.2.3.2.2.3

Simplify the right side.

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Step 2.2.3.2.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 2.2.4

The final solution is all the values that make $x (- x + 4) \geq 0$ true.

$x = 0, 4$

Step 2.2.5

Use each root to create test intervals.

$x < 0$

$0 < x < 4$

$x > 4$

Step 2.2.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.2.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 2.2.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 2.2.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) (- (- 2) + 4) \geq 0$

Step 2.2.6.1.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.2.6.2

Test a value on the interval $0 < x < 4$ to see if it makes the inequality true.

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Step 2.2.6.2.1

Choose a value on the interval $0 < x < 4$ and see if this value makes the original inequality true.

$x = 2$

Step 2.2.6.2.2

Replace $x$ with $2$ in the original inequality.

$(2) (- (2) + 4) \geq 0$

Step 2.2.6.2.3

The left side $4$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.2.6.3

Test a value on the interval $x > 4$ to see if it makes the inequality true.

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Step 2.2.6.3.1

Choose a value on the interval $x > 4$ and see if this value makes the original inequality true.

$x = 6$

Step 2.2.6.3.2

Replace $x$ with $6$ in the original inequality.

$(6) (- (6) + 4) \geq 0$

Step 2.2.6.3.3

The left side $- 12$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.2.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

$x < 0$ False

$0 < x < 4$ True

$x > 4$ False

Step 2.2.7

The solution consists of all of the true intervals.

$0 \leq x \leq 4$

Step 2.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[0, 4]$

Set-Builder Notation:

${x | 0 \leq x \leq 4}$

Interval Notation:

$[0, 4]$

Set-Builder Notation:

${x | 0 \leq x \leq 4}$

Step 3

The expression is continuous.

Continuous

Step 4