Trigonometry Examples

$x - \sqrt{4 x - 3} = 4$

Step 1

Subtract $x$ from both sides of the equation.

$- \sqrt{4 x - 3} = 4 - x$

Step 2

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

${(- \sqrt{4 x - 3})}^{2} = {(4 - x)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x - 3}$ as ${(4 x - 3)}^{\frac{1}{2}}$ .

${(- {(4 x - 3)}^{\frac{1}{2}})}^{2} = {(4 - x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(4 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- {(4 x - 3)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(4 x - 3)}^{\frac{1}{2}})}^{2} = {(4 - x)}^{2}$

Step 3.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(4 x - 3)}^{\frac{1}{2}})}^{2} = {(4 - x)}^{2}$

Step 3.2.1.3

Multiply ${({(4 x - 3)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(4 x - 3)}^{\frac{1}{2}})}^{2} = {(4 - x)}^{2}$

Step 3.2.1.4

Multiply the exponents in ${({(4 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

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

${(4 x - 3)}^{\frac{1}{2} \cdot 2} = {(4 - x)}^{2}$

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

${(4 x - 3)}^{\frac{1}{2} \cdot 2} = {(4 - x)}^{2}$

Step 3.2.1.4.2.2

Rewrite the expression.

${(4 x - 3)}^{1} = {(4 - x)}^{2}$

Step 3.2.1.5

Simplify.

$4 x - 3 = {(4 - x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Rewrite ${(4 - x)}^{2}$ as $(4 - x) (4 - x)$ .

$4 x - 3 = (4 - x) (4 - x)$

Step 3.3.1.2

Expand $(4 - x) (4 - x)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$4 x - 3 = 4 (4 - x) - x (4 - x)$

Step 3.3.1.2.2

Apply the distributive property.

$4 x - 3 = 4 \cdot 4 + 4 (- x) - x (4 - x)$

Step 3.3.1.2.3

Apply the distributive property.

$4 x - 3 = 4 \cdot 4 + 4 (- x) - x \cdot 4 - x (- x)$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $4$ by $4$ .

$4 x - 3 = 16 + 4 (- x) - x \cdot 4 - x (- x)$

Step 3.3.1.3.1.2

Multiply $- 1$ by $4$ .

$4 x - 3 = 16 - 4 x - x \cdot 4 - x (- x)$

Step 3.3.1.3.1.3

Multiply $4$ by $- 1$ .

$4 x - 3 = 16 - 4 x - 4 x - x (- x)$

Step 3.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$4 x - 3 = 16 - 4 x - 4 x - 1 \cdot - 1 x \cdot x$

Step 3.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.3.1.5.1

Move $x$ .

$4 x - 3 = 16 - 4 x - 4 x - 1 \cdot - 1 (x \cdot x)$

Step 3.3.1.3.1.5.2

Multiply $x$ by $x$ .

$4 x - 3 = 16 - 4 x - 4 x - 1 \cdot - 1 x^{2}$

Step 3.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 x - 3 = 16 - 4 x - 4 x + 1 x^{2}$

Step 3.3.1.3.1.7

Multiply $x^{2}$ by $1$ .

$4 x - 3 = 16 - 4 x - 4 x + x^{2}$

Step 3.3.1.3.2

Subtract $4 x$ from $- 4 x$ .

$4 x - 3 = 16 - 8 x + x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$16 - 8 x + x^{2} = 4 x - 3$

Step 4.2

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

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Step 4.2.1

Subtract $4 x$ from both sides of the equation.

$16 - 8 x + x^{2} - 4 x = - 3$

Step 4.2.2

Subtract $4 x$ from $- 8 x$ .

$16 + x^{2} - 12 x = - 3$

Step 4.3

Move all terms to the left side of the equation and simplify.

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Step 4.3.1

Add $3$ to both sides of the equation.

$16 + x^{2} - 12 x + 3 = 0$

Step 4.3.2

Add $16$ and $3$ .

$x^{2} - 12 x + 19 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5

Substitute the values $a = 1$ , $b = - 12$ , and $c = 19$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 19)}}{2 \cdot 1}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 19}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot 19$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 19}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $19$ .

$x = \frac{12 \pm \sqrt{144 - 76}}{2 \cdot 1}$

Step 4.6.1.3

Subtract $76$ from $144$ .

$x = \frac{12 \pm \sqrt{68}}{2 \cdot 1}$

Step 4.6.1.4

Rewrite $68$ as $2^{2} \cdot 17$ .

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Step 4.6.1.4.1

Factor $4$ out of $68$ .

$x = \frac{12 \pm \sqrt{4 (17)}}{2 \cdot 1}$

Step 4.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{12 \pm \sqrt{2^{2} \cdot 17}}{2 \cdot 1}$

Step 4.6.1.5

Pull terms out from under the radical.

$x = \frac{12 \pm 2 \sqrt{17}}{2 \cdot 1}$

Step 4.6.2

Multiply $2$ by $1$ .

$x = \frac{12 \pm 2 \sqrt{17}}{2}$

Step 4.6.3

Simplify $\frac{12 \pm 2 \sqrt{17}}{2}$ .

$x = 6 \pm \sqrt{17}$

Step 4.7

The final answer is the combination of both solutions.

$x = 6 + \sqrt{17}, 6 - \sqrt{17}$

Step 5

Exclude the solutions that do not make $x - \sqrt{4 x - 3} = 4$ true.

$x = 6 + \sqrt{17}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 6 + \sqrt{17}$

Decimal Form:

$x = 10.12310562 \dots$