Basic Math Examples

$- \sqrt{y - 1} = y$

Step 1

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

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

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Apply the product rule to $- {(y - 1)}^{\frac{1}{2}}$ .

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

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

Tap for more steps...

Step 2.2.1.4.1

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

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

Step 2.2.1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.4.2.1

Cancel the common factor.

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

Step 2.2.1.4.2.2

Rewrite the expression.

${(y - 1)}^{1} = y^{2}$

Step 2.2.1.5

Simplify.

$y - 1 = y^{2}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

$y - 1 - y^{2} = 0$

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

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

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

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Simplify the numerator.

Tap for more steps...

Step 3.4.1.1

One to any power is one.

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 1}}{2 \cdot - 1}$

Step 3.4.1.2

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

Tap for more steps...

Step 3.4.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 1 \pm \sqrt{1 + 4 \cdot - 1}}{2 \cdot - 1}$

Step 3.4.1.2.2

Multiply $4$ by $- 1$ .

$y = \frac{- 1 \pm \sqrt{1 - 4}}{2 \cdot - 1}$

Step 3.4.1.3

Subtract $4$ from $1$ .

$y = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot - 1}$

Step 3.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$y = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot - 1}$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot - 1}$

Step 3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 1 \pm i \sqrt{3}}{2 \cdot - 1}$

Step 3.4.2

Multiply $2$ by $- 1$ .

$y = \frac{- 1 \pm i \sqrt{3}}{- 2}$

Step 3.4.3

Simplify $\frac{- 1 \pm i \sqrt{3}}{- 2}$ .

$y = \frac{1 \pm i \sqrt{3}}{2}$

Step 3.5

The final answer is the combination of both solutions.

$y = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$