Solve for a square root of a-3 = square root of a

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

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.

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

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

Simplify the right side.

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 3

Solve for $a$ .

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

Subtract $a$ from both sides of the equation.

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

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $a^{1}$ .

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

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

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

Multiply $- 3$ by $- 3$ .

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

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

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

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

Subtract $a$ from $a$ .

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

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

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

Subtract $9$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$36 a = 81$

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

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

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

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

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

Factor $9$ out of $81$ .

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

Cancel the common factors.

Factor $9$ out of $36$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

$No solution$

Solve for a square root of a-3 = square root of a

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