Algebra Examples

$\sqrt{a} - 3 = \sqrt{a}$

Step 1

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

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

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 $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.4.1

Cancel the common factor.

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

Step 2.2.1.4.2

Rewrite the expression.

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

Step 2.2.1.5

Simplify.

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

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

Subtract $a$ from both sides of the equation.

${(a^{\frac{1}{2}} - 3)}^{2} - a = 0$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Rewrite ${(a^{\frac{1}{2}} - 3)}^{2}$ as $(a^{\frac{1}{2}} - 3) (a^{\frac{1}{2}} - 3)$ .

$(a^{\frac{1}{2}} - 3) (a^{\frac{1}{2}} - 3) - a = 0$

Step 3.1.2.2

Expand $(a^{\frac{1}{2}} - 3) (a^{\frac{1}{2}} - 3)$ using the FOIL Method.

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Step 3.1.2.2.1

Apply the distributive property.

$a^{\frac{1}{2}} (a^{\frac{1}{2}} - 3) - 3 (a^{\frac{1}{2}} - 3) - a = 0$

Step 3.1.2.2.2

Apply the distributive property.

$a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot - 3 - 3 (a^{\frac{1}{2}} - 3) - a = 0$

Step 3.1.2.2.3

Apply the distributive property.

$a^{\frac{1}{2}} a^{\frac{1}{2}} + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3

Simplify and combine like terms.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Multiply $a^{\frac{1}{2}}$ by $a^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1.2.3.1.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a^{\frac{1}{2} + \frac{1}{2}} + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.1.2

Combine the numerators over the common denominator.

$a^{\frac{1 + 1}{2}} + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.1.3

Add $1$ and $1$ .

$a^{\frac{2}{2}} + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.1.4

Divide $2$ by $2$ .

$a^{1} + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.2

Simplify $a^{1}$ .

$a + a^{\frac{1}{2}} \cdot - 3 - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.3

Move $- 3$ to the left of $a^{\frac{1}{2}}$ .

$a - 3 \cdot a^{\frac{1}{2}} - 3 a^{\frac{1}{2}} - 3 \cdot - 3 - a = 0$

Step 3.1.2.3.1.4

Multiply $- 3$ by $- 3$ .

$a - 3 a^{\frac{1}{2}} - 3 a^{\frac{1}{2}} + 9 - a = 0$

Step 3.1.2.3.2

Subtract $3 a^{\frac{1}{2}}$ from $- 3 a^{\frac{1}{2}}$ .

$a - 6 a^{\frac{1}{2}} + 9 - a = 0$

Step 3.1.3

Combine the opposite terms in $a - 6 a^{\frac{1}{2}} + 9 - a$ .

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Step 3.1.3.1

Subtract $a$ from $a$ .

$- 6 a^{\frac{1}{2}} + 9 + 0 = 0$

Step 3.1.3.2

Add $- 6 a^{\frac{1}{2}} + 9$ and $0$ .

$- 6 a^{\frac{1}{2}} + 9 = 0$

Step 3.2

Subtract $9$ from both sides of the equation.

$- 6 a^{\frac{1}{2}} = - 9$

Step 3.3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(- 6 a^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${(- 6 a^{\frac{1}{2}})}^{2}$ .

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Step 3.4.1.1.1

Apply the product rule to $- 6 a^{\frac{1}{2}}$ .

${(- 6)}^{2} {(a^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 3.4.1.1.2

Raise $- 6$ to the power of $2$ .

$36 {(a^{\frac{1}{2}})}^{2} = {(- 9)}^{2}$

Step 3.4.1.1.3

Multiply the exponents in ${(a^{\frac{1}{2}})}^{2}$ .

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Step 3.4.1.1.3.1

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

$36 a^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 3.4.1.1.3.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.3.2.1

Cancel the common factor.

$36 a^{\frac{1}{2} \cdot 2} = {(- 9)}^{2}$

Step 3.4.1.1.3.2.2

Rewrite the expression.

$36 a^{1} = {(- 9)}^{2}$

Step 3.4.1.1.4

Simplify.

$36 a = {(- 9)}^{2}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Raise $- 9$ to the power of $2$ .

$36 a = 81$

Step 3.5

Divide each term in $36 a = 81$ by $36$ and simplify.

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Step 3.5.1

Divide each term in $36 a = 81$ by $36$ .

$\frac{36 a}{36} = \frac{81}{36}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $36$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{36 a}{36} = \frac{81}{36}$

Step 3.5.2.1.2

Divide $a$ by $1$ .

$a = \frac{81}{36}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Cancel the common factor of $81$ and $36$ .

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Step 3.5.3.1.1

Factor $9$ out of $81$ .

$a = \frac{9 (9)}{36}$

Step 3.5.3.1.2

Cancel the common factors.

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Step 3.5.3.1.2.1

Factor $9$ out of $36$ .

$a = \frac{9 \cdot 9}{9 \cdot 4}$

Step 3.5.3.1.2.2

Cancel the common factor.

$a = \frac{9 \cdot 9}{9 \cdot 4}$

Step 3.5.3.1.2.3

Rewrite the expression.

$a = \frac{9}{4}$

Step 4

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

$No solution$