Precalculus Examples

$x - \sqrt{10 x - 25} = 0$

Step 1

Factor $5$ out of $10 x - 25$ .

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Step 1.1

Factor $5$ out of $10 x$ .

$x - \sqrt{5 (2 x) - 25} = 0$

Step 1.2

Factor $5$ out of $- 25$ .

$x - \sqrt{5 (2 x) + 5 (- 5)} = 0$

Step 1.3

Factor $5$ out of $5 (2 x) + 5 (- 5)$ .

$x - \sqrt{5 (2 x - 5)} = 0$

Step 2

Subtract $x$ from both sides of the equation.

$- \sqrt{5 (2 x - 5)} = - x$

Step 3

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

${(- \sqrt{5 (2 x - 5)})}^{2} = {(- x)}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 (2 x - 5)}$ as ${(5 (2 x - 5))}^{\frac{1}{2}}$ .

${(- {(5 (2 x - 5))}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(- {(5 (2 x - 5))}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Apply the distributive property.

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

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

Multiply $2$ by $5$ .

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

Step 4.2.1.2.2

Multiply $5$ by $- 5$ .

${(- {(10 x - 25)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 4.2.1.2.3

Apply the product rule to $- {(10 x - 25)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(10 x - 25)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 4.2.1.2.4

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

$1 {({(10 x - 25)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 4.2.1.2.5

Multiply ${({(10 x - 25)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(10 x - 25)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Step 4.2.1.2.6

Multiply the exponents in ${({(10 x - 25)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.2.6.1

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

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

Step 4.2.1.2.6.2

Cancel the common factor of $2$ .

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Step 4.2.1.2.6.2.1

Cancel the common factor.

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

Step 4.2.1.2.6.2.2

Rewrite the expression.

${(10 x - 25)}^{1} = {(- x)}^{2}$

Step 4.2.1.3

Simplify.

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

Step 4.3

Simplify the right side.

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Step 4.3.1

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

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Step 4.3.1.1

Apply the product rule to $- x$ .

$10 x - 25 = {(- 1)}^{2} x^{2}$

Step 4.3.1.2

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

$10 x - 25 = 1 x^{2}$

Step 4.3.1.3

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

$10 x - 25 = x^{2}$

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

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

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Step 5.2.1.1

Reorder the expression.

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Step 5.2.1.1.1

Move $- 25$ .

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

Step 5.2.1.1.2

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

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

Step 5.2.1.2

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

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Rewrite $- 25$ as $- 1 (25)$ .

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

Step 5.2.1.5

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

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

Step 5.2.1.6

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

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

Step 5.2.2

Factor using the perfect square rule.

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Step 5.2.2.1

Rewrite $25$ as $5^{2}$ .

$- (x^{2} - 10 x + 5^{2}) = 0$

Step 5.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 5.2.2.3

Rewrite the polynomial.

$- (x^{2} - 2 \cdot x \cdot 5 + 5^{2}) = 0$

Step 5.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$- {(x - 5)}^{2} = 0$

Step 5.3

Divide each term in $- {(x - 5)}^{2} = 0$ by $- 1$ and simplify.

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Step 5.3.1

Divide each term in $- {(x - 5)}^{2} = 0$ by $- 1$ .

$\frac{- {(x - 5)}^{2}}{- 1} = \frac{0}{- 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Dividing two negative values results in a positive value.

$\frac{{(x - 5)}^{2}}{1} = \frac{0}{- 1}$

Step 5.3.2.2

Divide ${(x - 5)}^{2}$ by $1$ .

${(x - 5)}^{2} = \frac{0}{- 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Divide $0$ by $- 1$ .

${(x - 5)}^{2} = 0$

Step 5.4

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

$x - 5 = 0$

Step 5.5

Add $5$ to both sides of the equation.

$x = 5$