Algebra Examples

$5 x = \sqrt{8 + 10 x}$

Step 1

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

$\sqrt{8 + 10 x} = 5 x$

Step 2

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

${\sqrt{8 + 10 x}}^{2} = {(5 x)}^{2}$

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 $\sqrt{8 + 10 x}$ as ${(8 + 10 x)}^{\frac{1}{2}}$ .

${({(8 + 10 x)}^{\frac{1}{2}})}^{2} = {(5 x)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(8 + 10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(8 + 10 x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(8 + 10 x)}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

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.

${(8 + 10 x)}^{\frac{1}{2} \cdot 2} = {(5 x)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(8 + 10 x)}^{1} = {(5 x)}^{2}$

Step 3.2.1.2

Simplify.

$8 + 10 x = {(5 x)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(5 x)}^{2}$ .

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Step 3.3.1.1

Apply the product rule to $5 x$ .

$8 + 10 x = 5^{2} x^{2}$

Step 3.3.1.2

Raise $5$ to the power of $2$ .

$8 + 10 x = 25 x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $25 x^{2}$ from both sides of the equation.

$8 + 10 x - 25 x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $8 + 10 x - 25 x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $8$ .

$10 x - 25 x^{2} + 8 = 0$

Step 4.2.1.1.2

Reorder $10 x$ and $- 25 x^{2}$ .

$- 25 x^{2} + 10 x + 8 = 0$

Step 4.2.1.2

Factor $- 1$ out of $- 25 x^{2}$ .

$- (25 x^{2}) + 10 x + 8 = 0$

Step 4.2.1.3

Factor $- 1$ out of $10 x$ .

$- (25 x^{2}) - (- 10 x) + 8 = 0$

Step 4.2.1.4

Rewrite $8$ as $- 1 (- 8)$ .

$- (25 x^{2}) - (- 10 x) - 1 \cdot - 8 = 0$

Step 4.2.1.5

Factor $- 1$ out of $- (25 x^{2}) - (- 10 x)$ .

$- (25 x^{2} - 10 x) - 1 \cdot - 8 = 0$

Step 4.2.1.6

Factor $- 1$ out of $- (25 x^{2} - 10 x) - 1 (- 8)$ .

$- (25 x^{2} - 10 x - 8) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor by grouping.

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Step 4.2.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 25 \cdot - 8 = - 200$ and whose sum is $b = - 10$ .

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Step 4.2.2.1.1.1

Factor $- 10$ out of $- 10 x$ .

$- (25 x^{2} - 10 x - 8) = 0$

Step 4.2.2.1.1.2

Rewrite $- 10$ as $10$ plus $- 20$

$- (25 x^{2} + (10 - 20) x - 8) = 0$

Step 4.2.2.1.1.3

Apply the distributive property.

$- (25 x^{2} + 10 x - 20 x - 8) = 0$

Step 4.2.2.1.2

Factor out the greatest common factor from each group.

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Step 4.2.2.1.2.1

Group the first two terms and the last two terms.

$- ((25 x^{2} + 10 x) - 20 x - 8) = 0$

Step 4.2.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (5 x (5 x + 2) - 4 (5 x + 2)) = 0$

Step 4.2.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 2$ .

$- ((5 x + 2) (5 x - 4)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (5 x + 2) (5 x - 4) = 0$

Step 4.3

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

$5 x + 2 = 0$

$5 x - 4 = 0$

Step 4.4

Set $5 x + 2$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 4.4.2

Solve $5 x + 2 = 0$ for $x$ .

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Step 4.4.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 4.4.2.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 4.4.2.2.1

Divide each term in $5 x = - 2$ by $5$ .

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 2}{5}$

Step 4.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{5}$

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{2}{5}$

Step 4.5

Set $5 x - 4$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $5 x - 4$ equal to $0$ .

$5 x - 4 = 0$

Step 4.5.2

Solve $5 x - 4 = 0$ for $x$ .

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Step 4.5.2.1

Add $4$ to both sides of the equation.

$5 x = 4$

Step 4.5.2.2

Divide each term in $5 x = 4$ by $5$ and simplify.

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Step 4.5.2.2.1

Divide each term in $5 x = 4$ by $5$ .

$\frac{5 x}{5} = \frac{4}{5}$

Step 4.5.2.2.2

Simplify the left side.

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Step 4.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{4}{5}$

Step 4.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{5}$

Step 4.6

The final solution is all the values that make $- (5 x + 2) (5 x - 4) = 0$ true.

$x = - \frac{2}{5}, \frac{4}{5}$

Step 5

Exclude the solutions that do not make $5 x = \sqrt{8 + 10 x}$ true.

$x = \frac{4}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{5}$

Decimal Form:

$x = 0.8$