Solve for a square root of a+6 = square root of 2a-1

$\sqrt{a + 6} = \sqrt{2 a} - 1$

Step 1

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

${\sqrt{a + 6}}^{2} = {(\sqrt{2 a} - 1)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a + 6}$ as ${(a + 6)}^{\frac{1}{2}}$ .

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

Simplify the left side.

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

Multiply the exponents in ${({(a + 6)}^{\frac{1}{2}})}^{2}$ .

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

${(a + 6)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 a} - 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(a + 6)}^{\frac{1}{2} \cdot 2} = {(\sqrt{2 a} - 1)}^{2}$

Rewrite the expression.

${(a + 6)}^{1} = {(\sqrt{2 a} - 1)}^{2}$

Simplify.

$a + 6 = {(\sqrt{2 a} - 1)}^{2}$

Simplify the right side.

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

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

$a + 6 = (\sqrt{2 a} - 1) (\sqrt{2 a} - 1)$

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

Apply the distributive property.

$a + 6 = \sqrt{2 a} (\sqrt{2 a} - 1) - 1 (\sqrt{2 a} - 1)$

Apply the distributive property.

$a + 6 = \sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} \cdot - 1 - 1 (\sqrt{2 a} - 1)$

Apply the distributive property.

$a + 6 = \sqrt{2 a} \sqrt{2 a} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 a} \sqrt{2 a}$ .

Raise $\sqrt{2 a}$ to the power of $1$ .

$a + 6 = {\sqrt{2 a}}^{1} \sqrt{2 a} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Raise $\sqrt{2 a}$ to the power of $1$ .

$a + 6 = {\sqrt{2 a}}^{1} {\sqrt{2 a}}^{1} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

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

$a + 6 = {\sqrt{2 a}}^{1 + 1} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Add $1$ and $1$ .

$a + 6 = {\sqrt{2 a}}^{2} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

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

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

$a + 6 = {({(2 a)}^{\frac{1}{2}})}^{2} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

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

$a + 6 = {(2 a)}^{\frac{1}{2} \cdot 2} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

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

$a + 6 = {(2 a)}^{\frac{2}{2}} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$a + 6 = {(2 a)}^{\frac{2}{2}} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Rewrite the expression.

$a + 6 = {(2 a)}^{1} + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Simplify.

$a + 6 = 2 a + \sqrt{2 a} \cdot - 1 - 1 \sqrt{2 a} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{2 a}$ .

$a + 6 = 2 a - 1 \cdot \sqrt{2 a} - 1 \sqrt{2 a} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{2 a}$ as $- \sqrt{2 a}$ .

$a + 6 = 2 a - \sqrt{2 a} - 1 \sqrt{2 a} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{2 a}$ as $- \sqrt{2 a}$ .

$a + 6 = 2 a - \sqrt{2 a} - \sqrt{2 a} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$a + 6 = 2 a - \sqrt{2 a} - \sqrt{2 a} + 1$

Subtract $\sqrt{2 a}$ from $- \sqrt{2 a}$ .

$a + 6 = 2 a - 2 \sqrt{2 a} + 1$

Step 3

Solve for $- 2 \sqrt{2 a}$ .

Rewrite the equation as $2 a - 2 \sqrt{2 a} + 1 = a + 6$ .

$2 a - 2 \sqrt{2 a} + 1 = a + 6$

Move all terms not containing $- 2 \sqrt{2 a}$ to the right side of the equation.

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

$- 2 \sqrt{2 a} + 1 = a + 6 - 2 a$

Subtract $1$ from both sides of the equation.

$- 2 \sqrt{2 a} = a + 6 - 2 a - 1$

Subtract $2 a$ from $a$ .

$- 2 \sqrt{2 a} = - a + 6 - 1$

Subtract $1$ from $6$ .

$- 2 \sqrt{2 a} = - a + 5$

Step 4

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

${(- 2 \sqrt{2 a})}^{2} = {(- a + 5)}^{2}$

Step 5

Simplify each side of the equation.

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

${(- 2 {(2 a)}^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Simplify the left side.

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

Apply the product rule to $2 a$ .

${(- 2 (2^{\frac{1}{2}} a^{\frac{1}{2}}))}^{2} = {(- a + 5)}^{2}$

Multiply $- 2 \cdot 2^{\frac{1}{2}}$ .

Factor out negative.

${(- (2 \cdot 2^{\frac{1}{2}}) a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Raise $2$ to the power of $1$ .

${(- (2^{1} \cdot 2^{\frac{1}{2}}) a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

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

${(- 2^{1 + \frac{1}{2}} a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Write $1$ as a fraction with a common denominator.

${(- 2^{\frac{2}{2} + \frac{1}{2}} a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Combine the numerators over the common denominator.

${(- 2^{\frac{2 + 1}{2}} a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Add $2$ and $1$ .

${(- 2^{\frac{3}{2}} a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- 2^{\frac{3}{2}} a^{\frac{1}{2}}$ .

${(- 2^{\frac{3}{2}})}^{2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Apply the product rule to $- 2^{\frac{3}{2}}$ .

${(- 1)}^{2} {(2^{\frac{3}{2}})}^{2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {(2^{\frac{3}{2}})}^{2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Multiply ${(2^{\frac{3}{2}})}^{2}$ by $1$ .

${(2^{\frac{3}{2}})}^{2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

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

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

$2^{\frac{3}{2} \cdot 2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2^{\frac{3}{2} \cdot 2} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

Rewrite the expression.

$2^{3} {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

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

$8 {(a^{\frac{1}{2}})}^{2} = {(- a + 5)}^{2}$

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

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

$8 a^{\frac{1}{2} \cdot 2} = {(- a + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$8 a^{\frac{1}{2} \cdot 2} = {(- a + 5)}^{2}$

Rewrite the expression.

$8 a^{1} = {(- a + 5)}^{2}$

Simplify.

$8 a = {(- a + 5)}^{2}$

Simplify the right side.

Simplify ${(- a + 5)}^{2}$ .

Rewrite ${(- a + 5)}^{2}$ as $(- a + 5) (- a + 5)$ .

$8 a = (- a + 5) (- a + 5)$

Expand $(- a + 5) (- a + 5)$ using the FOIL Method.

Apply the distributive property.

$8 a = - a (- a + 5) + 5 (- a + 5)$

Apply the distributive property.

$8 a = - a (- a) - a \cdot 5 + 5 (- a + 5)$

Apply the distributive property.

$8 a = - a (- a) - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 a = - 1 \cdot - 1 a \cdot a - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$8 a = - 1 \cdot - 1 (a \cdot a) - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Multiply $a$ by $a$ .

$8 a = - 1 \cdot - 1 a^{2} - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Multiply $- 1$ by $- 1$ .

$8 a = 1 a^{2} - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Multiply $a^{2}$ by $1$ .

$8 a = a^{2} - a \cdot 5 + 5 (- a) + 5 \cdot 5$

Multiply $5$ by $- 1$ .

$8 a = a^{2} - 5 a + 5 (- a) + 5 \cdot 5$

Multiply $- 1$ by $5$ .

$8 a = a^{2} - 5 a - 5 a + 5 \cdot 5$

Multiply $5$ by $5$ .

$8 a = a^{2} - 5 a - 5 a + 25$

Subtract $5 a$ from $- 5 a$ .

$8 a = a^{2} - 10 a + 25$

Step 6

Solve for $a$ .

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

$a^{2} - 10 a + 25 = 8 a$

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

Subtract $8 a$ from both sides of the equation.

$a^{2} - 10 a + 25 - 8 a = 0$

Subtract $8 a$ from $- 10 a$ .

$a^{2} - 18 a + 25 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 18$ , and $c = 25$ into the quadratic formula and solve for $a$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 18$ to the power of $2$ .

$a = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$a = \frac{18 \pm \sqrt{324 - 4 \cdot 25}}{2 \cdot 1}$

Multiply $- 4$ by $25$ .

$a = \frac{18 \pm \sqrt{324 - 100}}{2 \cdot 1}$

Subtract $100$ from $324$ .

$a = \frac{18 \pm \sqrt{224}}{2 \cdot 1}$

Rewrite $224$ as $4^{2} \cdot 14$ .

Factor $16$ out of $224$ .

$a = \frac{18 \pm \sqrt{16 (14)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$a = \frac{18 \pm \sqrt{4^{2} \cdot 14}}{2 \cdot 1}$

Pull terms out from under the radical.

$a = \frac{18 \pm 4 \sqrt{14}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$a = \frac{18 \pm 4 \sqrt{14}}{2}$

Simplify $\frac{18 \pm 4 \sqrt{14}}{2}$ .

$a = 9 \pm 2 \sqrt{14}$

The final answer is the combination of both solutions.

$a = 9 + 2 \sqrt{14}, 9 - 2 \sqrt{14}$

Step 7

Exclude the solutions that do not make $\sqrt{a + 6} = \sqrt{2 a} - 1$ true.

$a = 9 + 2 \sqrt{14}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = 9 + 2 \sqrt{14}$

Decimal Form:

$a = 16.48331477 \dots$

Solve for a square root of a+6 = square root of 2a-1

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