Solve for s square root of s^2-1 = square root of 5s-5

$\sqrt{s^{2} - 1} = \sqrt{5 s - 5}$

Step 1

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

${\sqrt{s^{2} - 1}}^{2} = {\sqrt{5 s - 5}}^{2}$

Step 2

Simplify each side of the equation.

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

${({(s^{2} - 1)}^{\frac{1}{2}})}^{2} = {\sqrt{5 s - 5}}^{2}$

Simplify the left side.

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

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

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

${(s^{2} - 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 s - 5}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(s^{2} - 1)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 s - 5}}^{2}$

Rewrite the expression.

${(s^{2} - 1)}^{1} = {\sqrt{5 s - 5}}^{2}$

Simplify.

$s^{2} - 1 = {\sqrt{5 s - 5}}^{2}$

Simplify the right side.

Simplify ${\sqrt{5 s - 5}}^{2}$ .

Factor $5$ out of $5 s - 5$ .

Factor $5$ out of $5 s$ .

$s^{2} - 1 = {\sqrt{5 (s) - 5}}^{2}$

Factor $5$ out of $- 5$ .

$s^{2} - 1 = {\sqrt{5 (s) + 5 (- 1)}}^{2}$

Factor $5$ out of $5 (s) + 5 (- 1)$ .

$s^{2} - 1 = {\sqrt{5 (s - 1)}}^{2}$

Rewrite ${\sqrt{5 (s - 1)}}^{2}$ as $5 (s - 1)$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 (s - 1)}$ as ${(5 (s - 1))}^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$s^{2} - 1 = {(5 (s - 1))}^{1}$

Simplify.

$s^{2} - 1 = 5 (s - 1)$

Apply the distributive property.

$s^{2} - 1 = 5 s + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$s^{2} - 1 = 5 s - 5$

Step 3

Solve for $s$ .

Subtract $5 s$ from both sides of the equation.

$s^{2} - 1 - 5 s = - 5$

Add $5$ to both sides of the equation.

$s^{2} - 1 - 5 s + 5 = 0$

Add $- 1$ and $5$ .

$s^{2} - 5 s + 4 = 0$

Factor $s^{2} - 5 s + 4$ 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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Write the factored form using these integers.

$(s - 4) (s - 1) = 0$

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

$s - 4 = 0$

$s - 1 = 0$

Set $s - 4$ equal to $0$ and solve for $s$ .

Set $s - 4$ equal to $0$ .

$s - 4 = 0$

Add $4$ to both sides of the equation.

$s = 4$

Set $s - 1$ equal to $0$ and solve for $s$ .

Set $s - 1$ equal to $0$ .

$s - 1 = 0$

Add $1$ to both sides of the equation.

$s = 1$

The final solution is all the values that make $(s - 4) (s - 1) = 0$ true.

$s = 4, 1$

Solve for s square root of s^2-1 = square root of 5s-5

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