Solve for s square root of 3-s=s-1

$\sqrt{3 - s} = s - 1$

Step 1

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

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

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - s}$ as ${(3 - s)}^{\frac{1}{2}}$ .

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

Simplify the left side.

Simplify ${({(3 - s)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 - s)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

${(3 - s)}^{1} = {(s - 1)}^{2}$

Simplify.

$3 - s = {(s - 1)}^{2}$

Simplify the right side.

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

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

$3 - s = (s - 1) (s - 1)$

Expand $(s - 1) (s - 1)$ using the FOIL Method.

Apply the distributive property.

$3 - s = s (s - 1) - 1 (s - 1)$

Apply the distributive property.

$3 - s = s \cdot s + s \cdot - 1 - 1 (s - 1)$

Apply the distributive property.

$3 - s = s \cdot s + s \cdot - 1 - 1 s - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $s$ by $s$ .

$3 - s = s^{2} + s \cdot - 1 - 1 s - 1 \cdot - 1$

Move $- 1$ to the left of $s$ .

$3 - s = s^{2} - 1 \cdot s - 1 s - 1 \cdot - 1$

Rewrite $- 1 s$ as $- s$ .

$3 - s = s^{2} - s - 1 s - 1 \cdot - 1$

Rewrite $- 1 s$ as $- s$ .

$3 - s = s^{2} - s - s - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$3 - s = s^{2} - s - s + 1$

Subtract $s$ from $- s$ .

$3 - s = s^{2} - 2 s + 1$

Step 3

Solve for $s$ .

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

$s^{2} - 2 s + 1 = 3 - s$

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

Add $s$ to both sides of the equation.

$s^{2} - 2 s + 1 + s = 3$

Add $- 2 s$ and $s$ .

$s^{2} - s + 1 = 3$

Subtract $3$ from both sides of the equation.

$s^{2} - s + 1 - 3 = 0$

Subtract $3$ from $1$ .

$s^{2} - s - 2 = 0$

Factor $s^{2} - s - 2$ 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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

$(s - 2) (s + 1) = 0$

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

$s - 2 = 0$

$s + 1 = 0$

Set $s - 2$ equal to $0$ and solve for $s$ .

Set $s - 2$ equal to $0$ .

$s - 2 = 0$

Add $2$ to both sides of the equation.

$s = 2$

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

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

$s + 1 = 0$

Subtract $1$ from both sides of the equation.

$s = - 1$

The final solution is all the values that make $(s - 2) (s + 1) = 0$ true.

$s = 2, - 1$

Step 4

Exclude the solutions that do not make $\sqrt{3 - s} = s - 1$ true.

$s = 2$

Solve for s square root of 3-s=s-1

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