Solve for a a = square root of 8a-15

$a = \sqrt{8 a - 15}$

Step 1

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

$\sqrt{8 a - 15} = a$

Step 2

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

${\sqrt{8 a - 15}}^{2} = a^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 a - 15}$ as ${(8 a - 15)}^{\frac{1}{2}}$ .

${({(8 a - 15)}^{\frac{1}{2}})}^{2} = a^{2}$

Simplify the left side.

Simplify ${({(8 a - 15)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8 a - 15)}^{\frac{1}{2}})}^{2}$ .

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

${(8 a - 15)}^{\frac{1}{2} \cdot 2} = a^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 a - 15)}^{\frac{1}{2} \cdot 2} = a^{2}$

Rewrite the expression.

${(8 a - 15)}^{1} = a^{2}$

Simplify.

$8 a - 15 = a^{2}$

Step 4

Solve for $a$ .

Subtract $a^{2}$ from both sides of the equation.

$8 a - 15 - a^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $8 a - 15 - a^{2}$ .

Reorder the expression.

Move $- 15$ .

$8 a - a^{2} - 15 = 0$

Reorder $8 a$ and $- a^{2}$ .

$- a^{2} + 8 a - 15 = 0$

Factor $- 1$ out of $- a^{2}$ .

$- (a^{2}) + 8 a - 15 = 0$

Factor $- 1$ out of $8 a$ .

$- (a^{2}) - (- 8 a) - 15 = 0$

Rewrite $- 15$ as $- 1 (15)$ .

$- (a^{2}) - (- 8 a) - 1 \cdot 15 = 0$

Factor $- 1$ out of $- (a^{2}) - (- 8 a)$ .

$- (a^{2} - 8 a) - 1 \cdot 15 = 0$

Factor $- 1$ out of $- (a^{2} - 8 a) - 1 (15)$ .

$- (a^{2} - 8 a + 15) = 0$

Factor.

Factor $a^{2} - 8 a + 15$ using the AC method.

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$- ((a - 5) (a - 3)) = 0$

Remove unnecessary parentheses.

$- (a - 5) (a - 3) = 0$

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

$a - 5 = 0$

$a - 3 = 0$

Set $a - 5$ equal to $0$ and solve for $a$ .

Set $a - 5$ equal to $0$ .

$a - 5 = 0$

Add $5$ to both sides of the equation.

$a = 5$

Set $a - 3$ equal to $0$ and solve for $a$ .

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Add $3$ to both sides of the equation.

$a = 3$

The final solution is all the values that make $- (a - 5) (a - 3) = 0$ true.

$a = 5, 3$

Solve for a a = square root of 8a-15

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