Algebra Examples

$\sqrt{x^{2} - 10 x + 25} + 12 \sqrt{x} = 15 \sqrt{x}$

Step 1

Move all terms not containing $\sqrt{x^{2} - 10 x + 25}$ to the right side of the equation.

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Step 1.1

Subtract $12 \sqrt{x}$ from both sides of the equation.

$\sqrt{x^{2} - 10 x + 25} = 15 \sqrt{x} - 12 \sqrt{x}$

Step 1.2

Subtract $12 \sqrt{x}$ from $15 \sqrt{x}$ .

$\sqrt{x^{2} - 10 x + 25} = 3 \sqrt{x}$

Step 2

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

${\sqrt{x^{2} - 10 x + 25}}^{2} = {(3 \sqrt{x})}^{2}$

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 $\sqrt{x^{2} - 10 x + 25}$ as ${(x^{2} - 10 x + 25)}^{\frac{1}{2}}$ .

${({(x^{2} - 10 x + 25)}^{\frac{1}{2}})}^{2} = {(3 \sqrt{x})}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x^{2} - 10 x + 25)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

${(x^{2} - 10 x + 25)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x})}^{2}$

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.

${(x^{2} - 10 x + 25)}^{\frac{1}{2} \cdot 2} = {(3 \sqrt{x})}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 10 x + 25)}^{1} = {(3 \sqrt{x})}^{2}$

Step 3.2.1.2

Simplify.

$x^{2} - 10 x + 25 = {(3 \sqrt{x})}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(3 \sqrt{x})}^{2}$ .

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Step 3.3.1.1

Simplify the expression.

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Step 3.3.1.1.1

Apply the product rule to $3 \sqrt{x}$ .

$x^{2} - 10 x + 25 = 3^{2} {\sqrt{x}}^{2}$

Step 3.3.1.1.2

Raise $3$ to the power of $2$ .

$x^{2} - 10 x + 25 = 9 {\sqrt{x}}^{2}$

Step 3.3.1.2

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

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Step 3.3.1.2.1

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

$x^{2} - 10 x + 25 = 9 {(x^{\frac{1}{2}})}^{2}$

Step 3.3.1.2.2

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

$x^{2} - 10 x + 25 = 9 x^{\frac{1}{2} \cdot 2}$

Step 3.3.1.2.3

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

$x^{2} - 10 x + 25 = 9 x^{\frac{2}{2}}$

Step 3.3.1.2.4

Cancel the common factor of $2$ .

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Step 3.3.1.2.4.1

Cancel the common factor.

$x^{2} - 10 x + 25 = 9 x^{\frac{2}{2}}$

Step 3.3.1.2.4.2

Rewrite the expression.

$x^{2} - 10 x + 25 = 9 x^{1}$

Step 3.3.1.2.5

Simplify.

$x^{2} - 10 x + 25 = 9 x$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $9 x$ from both sides of the equation.

$x^{2} - 10 x + 25 - 9 x = 0$

Step 4.1.2

Subtract $9 x$ from $- 10 x$ .

$x^{2} - 19 x + 25 = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = 1$ , $b = - 19$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 25}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{19 \pm \sqrt{361 - 100}}{2 \cdot 1}$

Step 4.4.1.3

Subtract $100$ from $361$ .

$x = \frac{19 \pm \sqrt{261}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $261$ as $3^{2} \cdot 29$ .

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Step 4.4.1.4.1

Factor $9$ out of $261$ .

$x = \frac{19 \pm \sqrt{9 (29)}}{2 \cdot 1}$

Step 4.4.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{19 \pm \sqrt{3^{2} \cdot 29}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{19 \pm 3 \sqrt{29}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{19 \pm 3 \sqrt{29}}{2}$

Step 4.5

The final answer is the combination of both solutions.

$x = \frac{19 + 3 \sqrt{29}}{2}, \frac{19 - 3 \sqrt{29}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19 + 3 \sqrt{29}}{2}, \frac{19 - 3 \sqrt{29}}{2}$

Decimal Form:

$x = 17.57774721 \dots, 1.42225278 \dots$