Algebra Examples

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

Step 1

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

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

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$2 x - 3 = (x - 3) (x - 3)$

Step 2.3.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$2 x - 3 = x (x - 3) - 3 (x - 3)$

Step 2.3.1.2.2

Apply the distributive property.

$2 x - 3 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Step 2.3.1.2.3

Apply the distributive property.

$2 x - 3 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$2 x - 3 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Step 2.3.1.3.1.2

Move $- 3$ to the left of $x$ .

$2 x - 3 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Step 2.3.1.3.1.3

Multiply $- 3$ by $- 3$ .

$2 x - 3 = x^{2} - 3 x - 3 x + 9$

Step 2.3.1.3.2

Subtract $3 x$ from $- 3 x$ .

$2 x - 3 = x^{2} - 6 x + 9$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} - 6 x + 9 = 2 x - 3$

Step 3.2

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

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Step 3.2.1

Subtract $2 x$ from both sides of the equation.

$x^{2} - 6 x + 9 - 2 x = - 3$

Step 3.2.2

Subtract $2 x$ from $- 6 x$ .

$x^{2} - 8 x + 9 = - 3$

Step 3.3

Add $3$ to both sides of the equation.

$x^{2} - 8 x + 9 + 3 = 0$

Step 3.4

Add $9$ and $3$ .

$x^{2} - 8 x + 12 = 0$

Step 3.5

Factor $x^{2} - 8 x + 12$ using the AC method.

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Step 3.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 3.5.2

Write the factored form using these integers.

$(x - 6) (x - 2) = 0$

Step 3.6

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

$x - 6 = 0$

$x - 2 = 0$

Step 3.7

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.7.2

Add $6$ to both sides of the equation.

$x = 6$

Step 3.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.9

The final solution is all the values that make $(x - 6) (x - 2) = 0$ true.

$x = 6, 2$

Step 4

Exclude the solutions that do not make $\sqrt{2 x - 3} = x - 3$ true.

$x = 6$