Algebra Examples

${(8 - \sqrt{x})}^{2} = {(4 + \sqrt{x})}^{2}$

Step 1

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

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Step 1.1

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

$(8 - \sqrt{x}) (8 - \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$8 (8 - \sqrt{x}) - \sqrt{x} (8 - \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.2.2

Apply the distributive property.

$8 \cdot 8 + 8 (- \sqrt{x}) - \sqrt{x} (8 - \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.2.3

Apply the distributive property.

$8 \cdot 8 + 8 (- \sqrt{x}) - \sqrt{x} \cdot 8 - \sqrt{x} (- \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $8$ by $8$ .

$64 + 8 (- \sqrt{x}) - \sqrt{x} \cdot 8 - \sqrt{x} (- \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.2

Multiply $- 1$ by $8$ .

$64 - 8 \sqrt{x} - \sqrt{x} \cdot 8 - \sqrt{x} (- \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.3

Multiply $8$ by $- 1$ .

$64 - 8 \sqrt{x} - 8 \sqrt{x} - \sqrt{x} (- \sqrt{x}) = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4

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

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Step 1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$64 - 8 \sqrt{x} - 8 \sqrt{x} + 1 \sqrt{x} \sqrt{x} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4.2

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + \sqrt{x} \sqrt{x} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4.3

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4.4

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4.5

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + {\sqrt{x}}^{1 + 1} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.4.6

Add $1$ and $1$ .

$64 - 8 \sqrt{x} - 8 \sqrt{x} + {\sqrt{x}}^{2} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5

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

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Step 1.3.1.5.1

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + {(x^{\frac{1}{2}})}^{2} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5.2

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + x^{\frac{1}{2} \cdot 2} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5.3

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

$64 - 8 \sqrt{x} - 8 \sqrt{x} + x^{\frac{2}{2}} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5.4

Cancel the common factor of $2$ .

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Step 1.3.1.5.4.1

Cancel the common factor.

$64 - 8 \sqrt{x} - 8 \sqrt{x} + x^{\frac{2}{2}} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5.4.2

Rewrite the expression.

$64 - 8 \sqrt{x} - 8 \sqrt{x} + x^{1} = {(4 + \sqrt{x})}^{2}$

Step 1.3.1.5.5

Simplify.

$64 - 8 \sqrt{x} - 8 \sqrt{x} + x = {(4 + \sqrt{x})}^{2}$

Step 1.3.2

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

$64 - 16 \sqrt{x} + x = {(4 + \sqrt{x})}^{2}$

Step 2

Simplify ${(4 + \sqrt{x})}^{2}$ .

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Step 2.1

Rewrite ${(4 + \sqrt{x})}^{2}$ as $(4 + \sqrt{x}) (4 + \sqrt{x})$ .

$64 - 16 \sqrt{x} + x = (4 + \sqrt{x}) (4 + \sqrt{x})$

Step 2.2

Expand $(4 + \sqrt{x}) (4 + \sqrt{x})$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$64 - 16 \sqrt{x} + x = 4 (4 + \sqrt{x}) + \sqrt{x} (4 + \sqrt{x})$

Step 2.2.2

Apply the distributive property.

$64 - 16 \sqrt{x} + x = 4 \cdot 4 + 4 \sqrt{x} + \sqrt{x} (4 + \sqrt{x})$

Step 2.2.3

Apply the distributive property.

$64 - 16 \sqrt{x} + x = 4 \cdot 4 + 4 \sqrt{x} + \sqrt{x} \cdot 4 + \sqrt{x} \sqrt{x}$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $4$ by $4$ .

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + \sqrt{x} \cdot 4 + \sqrt{x} \sqrt{x}$

Step 2.3.1.2

Move $4$ to the left of $\sqrt{x}$ .

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \cdot \sqrt{x} + \sqrt{x} \sqrt{x}$

Step 2.3.1.3

Multiply $\sqrt{x} \sqrt{x}$ .

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Step 2.3.1.3.1

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + {\sqrt{x}}^{1} \sqrt{x}$

Step 2.3.1.3.2

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + {\sqrt{x}}^{1} {\sqrt{x}}^{1}$

Step 2.3.1.3.3

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + {\sqrt{x}}^{1 + 1}$

Step 2.3.1.3.4

Add $1$ and $1$ .

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + {\sqrt{x}}^{2}$

Step 2.3.1.4

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

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Step 2.3.1.4.1

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + {(x^{\frac{1}{2}})}^{2}$

Step 2.3.1.4.2

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + x^{\frac{1}{2} \cdot 2}$

Step 2.3.1.4.3

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

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.4.1

Cancel the common factor.

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + x^{\frac{2}{2}}$

Step 2.3.1.4.4.2

Rewrite the expression.

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + x^{1}$

Step 2.3.1.4.5

Simplify.

$64 - 16 \sqrt{x} + x = 16 + 4 \sqrt{x} + 4 \sqrt{x} + x$

Step 2.3.2

Add $4 \sqrt{x}$ and $4 \sqrt{x}$ .

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

Step 3

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

$x = 4$

Step 4