Algebra Examples

$\sqrt{2 x - 3} = \sqrt{10 - x}$

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 3}$ as ${(2 x - 3)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Rewrite ${\sqrt{10 - x}}^{2}$ as $10 - x$ .

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Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 - x}$ as ${(10 - x)}^{\frac{1}{2}}$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

$2 x - 3 = {(10 - x)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

$2 x - 3 = {(10 - x)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$2 x - 3 = {(10 - x)}^{1}$

Step 2.3.1.5

Simplify.

$2 x - 3 = 10 - x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Add $x$ to both sides of the equation.

$2 x - 3 + x = 10$

Step 3.1.2

Add $2 x$ and $x$ .

$3 x - 3 = 10$

Step 3.2

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

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Step 3.2.1

Add $3$ to both sides of the equation.

$3 x = 10 + 3$

Step 3.2.2

Add $10$ and $3$ .

$3 x = 13$

Step 3.3

Divide each term in $3 x = 13$ by $3$ and simplify.

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Step 3.3.1

Divide each term in $3 x = 13$ by $3$ .

$\frac{3 x}{3} = \frac{13}{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{13}{3}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{3}$

Decimal Form:

$x = 4.\overline{3}$

Mixed Number Form:

$x = 4 \frac{1}{3}$