Precalculus Examples

$\sqrt{16 + x} + \sqrt{16 - x} = 8$

Step 1

Subtract $\sqrt{16 - x}$ from both sides of the equation.

$\sqrt{16 + x} = 8 - \sqrt{16 - x}$

Step 2

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

${\sqrt{16 + x}}^{2} = {(8 - \sqrt{16 - x})}^{2}$

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 $\sqrt{16 + x}$ as ${(16 + x)}^{\frac{1}{2}}$ .

${({(16 + x)}^{\frac{1}{2}})}^{2} = {(8 - \sqrt{16 - x})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(16 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(16 + x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(16 + x)}^{\frac{1}{2} \cdot 2} = {(8 - \sqrt{16 - x})}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(16 + x)}^{\frac{1}{2} \cdot 2} = {(8 - \sqrt{16 - x})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(16 + x)}^{1} = {(8 - \sqrt{16 - x})}^{2}$

Step 3.2.1.2

Simplify.

$16 + x = {(8 - \sqrt{16 - x})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(8 - \sqrt{16 - x})}^{2}$ .

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Step 3.3.1.1

Rewrite ${(8 - \sqrt{16 - x})}^{2}$ as $(8 - \sqrt{16 - x}) (8 - \sqrt{16 - x})$ .

$16 + x = (8 - \sqrt{16 - x}) (8 - \sqrt{16 - x})$

Step 3.3.1.2

Expand $(8 - \sqrt{16 - x}) (8 - \sqrt{16 - x})$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$16 + x = 8 (8 - \sqrt{16 - x}) - \sqrt{16 - x} (8 - \sqrt{16 - x})$

Step 3.3.1.2.2

Apply the distributive property.

$16 + x = 8 \cdot 8 + 8 (- \sqrt{16 - x}) - \sqrt{16 - x} (8 - \sqrt{16 - x})$

Step 3.3.1.2.3

Apply the distributive property.

$16 + x = 8 \cdot 8 + 8 (- \sqrt{16 - x}) - \sqrt{16 - x} \cdot 8 - \sqrt{16 - x} (- \sqrt{16 - x})$

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 $8$ by $8$ .

$16 + x = 64 + 8 (- \sqrt{16 - x}) - \sqrt{16 - x} \cdot 8 - \sqrt{16 - x} (- \sqrt{16 - x})$

Step 3.3.1.3.1.2

Multiply $- 1$ by $8$ .

$16 + x = 64 - 8 \sqrt{16 - x} - \sqrt{16 - x} \cdot 8 - \sqrt{16 - x} (- \sqrt{16 - x})$

Step 3.3.1.3.1.3

Multiply $8$ by $- 1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} - \sqrt{16 - x} (- \sqrt{16 - x})$

Step 3.3.1.3.1.4

Multiply $- \sqrt{16 - x} (- \sqrt{16 - x})$ .

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Step 3.3.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + 1 \sqrt{16 - x} \sqrt{16 - x}$

Step 3.3.1.3.1.4.2

Multiply $\sqrt{16 - x}$ by $1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + \sqrt{16 - x} \sqrt{16 - x}$

Step 3.3.1.3.1.4.3

Raise $\sqrt{16 - x}$ to the power of $1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {\sqrt{16 - x}}^{1} \sqrt{16 - x}$

Step 3.3.1.3.1.4.4

Raise $\sqrt{16 - x}$ to the power of $1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {\sqrt{16 - x}}^{1} {\sqrt{16 - x}}^{1}$

Step 3.3.1.3.1.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {\sqrt{16 - x}}^{1 + 1}$

Step 3.3.1.3.1.4.6

Add $1$ and $1$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {\sqrt{16 - x}}^{2}$

Step 3.3.1.3.1.5

Rewrite ${\sqrt{16 - x}}^{2}$ as $16 - x$ .

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Step 3.3.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 - x}$ as ${(16 - x)}^{\frac{1}{2}}$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {({(16 - x)}^{\frac{1}{2}})}^{2}$

Step 3.3.1.3.1.5.2

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

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {(16 - x)}^{\frac{1}{2} \cdot 2}$

Step 3.3.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {(16 - x)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.1.3.1.5.4.1

Cancel the common factor.

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {(16 - x)}^{\frac{2}{2}}$

Step 3.3.1.3.1.5.4.2

Rewrite the expression.

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + {(16 - x)}^{1}$

Step 3.3.1.3.1.5.5

Simplify.

$16 + x = 64 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} + 16 - x$

Step 3.3.1.3.2

Add $64$ and $16$ .

$16 + x = 80 - 8 \sqrt{16 - x} - 8 \sqrt{16 - x} - x$

Step 3.3.1.3.3

Subtract $8 \sqrt{16 - x}$ from $- 8 \sqrt{16 - x}$ .

$16 + x = 80 - 16 \sqrt{16 - x} - x$

Step 4

Solve for $- 16 \sqrt{16 - x}$ .

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Step 4.1

Rewrite the equation as $80 - 16 \sqrt{16 - x} - x = 16 + x$ .

$80 - 16 \sqrt{16 - x} - x = 16 + x$

Step 4.2

Move all terms not containing $- 16 \sqrt{16 - x}$ to the right side of the equation.

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Step 4.2.1

Subtract $80$ from both sides of the equation.

$- 16 \sqrt{16 - x} - x = 16 + x - 80$

Step 4.2.2

Add $x$ to both sides of the equation.

$- 16 \sqrt{16 - x} = 16 + x - 80 + x$

Step 4.2.3

Subtract $80$ from $16$ .

$- 16 \sqrt{16 - x} = x - 64 + x$

Step 4.2.4

Add $x$ and $x$ .

$- 16 \sqrt{16 - x} = 2 x - 64$

Step 5

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

${(- 16 \sqrt{16 - x})}^{2} = {(2 x - 64)}^{2}$

Step 6

Simplify each side of the equation.

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Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{16 - x}$ as ${(16 - x)}^{\frac{1}{2}}$ .

${(- 16 {(16 - x)}^{\frac{1}{2}})}^{2} = {(2 x - 64)}^{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Simplify ${(- 16 {(16 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.1

Apply the product rule to $- 16 {(16 - x)}^{\frac{1}{2}}$ .

${(- 16)}^{2} {({(16 - x)}^{\frac{1}{2}})}^{2} = {(2 x - 64)}^{2}$

Step 6.2.1.2

Raise $- 16$ to the power of $2$ .

$256 {({(16 - x)}^{\frac{1}{2}})}^{2} = {(2 x - 64)}^{2}$

Step 6.2.1.3

Multiply the exponents in ${({(16 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.1.3.1

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

$256 {(16 - x)}^{\frac{1}{2} \cdot 2} = {(2 x - 64)}^{2}$

Step 6.2.1.3.2

Cancel the common factor of $2$ .

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Step 6.2.1.3.2.1

Cancel the common factor.

$256 {(16 - x)}^{\frac{1}{2} \cdot 2} = {(2 x - 64)}^{2}$

Step 6.2.1.3.2.2

Rewrite the expression.

$256 {(16 - x)}^{1} = {(2 x - 64)}^{2}$

Step 6.2.1.4

Simplify.

$256 (16 - x) = {(2 x - 64)}^{2}$

Step 6.2.1.5

Apply the distributive property.

$256 \cdot 16 + 256 (- x) = {(2 x - 64)}^{2}$

Step 6.2.1.6

Multiply.

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Step 6.2.1.6.1

Multiply $256$ by $16$ .

$4096 + 256 (- x) = {(2 x - 64)}^{2}$

Step 6.2.1.6.2

Multiply $- 1$ by $256$ .

$4096 - 256 x = {(2 x - 64)}^{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify ${(2 x - 64)}^{2}$ .

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Step 6.3.1.1

Rewrite ${(2 x - 64)}^{2}$ as $(2 x - 64) (2 x - 64)$ .

$4096 - 256 x = (2 x - 64) (2 x - 64)$

Step 6.3.1.2

Expand $(2 x - 64) (2 x - 64)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$4096 - 256 x = 2 x (2 x - 64) - 64 (2 x - 64)$

Step 6.3.1.2.2

Apply the distributive property.

$4096 - 256 x = 2 x (2 x) + 2 x \cdot - 64 - 64 (2 x - 64)$

Step 6.3.1.2.3

Apply the distributive property.

$4096 - 256 x = 2 x (2 x) + 2 x \cdot - 64 - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$4096 - 256 x = 2 \cdot 2 x \cdot x + 2 x \cdot - 64 - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3.1.2

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

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Step 6.3.1.3.1.2.1

Move $x$ .

$4096 - 256 x = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 64 - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3.1.2.2

Multiply $x$ by $x$ .

$4096 - 256 x = 2 \cdot 2 x^{2} + 2 x \cdot - 64 - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3.1.3

Multiply $2$ by $2$ .

$4096 - 256 x = 4 x^{2} + 2 x \cdot - 64 - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3.1.4

Multiply $- 64$ by $2$ .

$4096 - 256 x = 4 x^{2} - 128 x - 64 (2 x) - 64 \cdot - 64$

Step 6.3.1.3.1.5

Multiply $2$ by $- 64$ .

$4096 - 256 x = 4 x^{2} - 128 x - 128 x - 64 \cdot - 64$

Step 6.3.1.3.1.6

Multiply $- 64$ by $- 64$ .

$4096 - 256 x = 4 x^{2} - 128 x - 128 x + 4096$

Step 6.3.1.3.2

Subtract $128 x$ from $- 128 x$ .

$4096 - 256 x = 4 x^{2} - 256 x + 4096$

Step 7

Solve for $x$ .

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Step 7.1

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

$4 x^{2} - 256 x + 4096 = 4096 - 256 x$

Step 7.2

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

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Step 7.2.1

Add $256 x$ to both sides of the equation.

$4 x^{2} - 256 x + 4096 + 256 x = 4096$

Step 7.2.2

Combine the opposite terms in $4 x^{2} - 256 x + 4096 + 256 x$ .

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Step 7.2.2.1

Add $- 256 x$ and $256 x$ .

$4 x^{2} + 0 + 4096 = 4096$

Step 7.2.2.2

Add $4 x^{2}$ and $0$ .

$4 x^{2} + 4096 = 4096$

Step 7.3

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

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Step 7.3.1

Subtract $4096$ from both sides of the equation.

$4 x^{2} = 4096 - 4096$

Step 7.3.2

Subtract $4096$ from $4096$ .

$4 x^{2} = 0$

Step 7.4

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 7.4.1

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $4$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 7.4.2.1.2

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

$x^{2} = \frac{0}{4}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 7.5

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

$x = \pm \sqrt{0}$

Step 7.6

Simplify $\pm \sqrt{0}$ .

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Step 7.6.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 7.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 7.6.3

Plus or minus $0$ is $0$ .

$x = 0$