Algebra Examples

$\sqrt{9 y^{2} + 15 y + 18} = \sqrt{5 y^{2} - 5 y - 7}$

Step 1

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

${\sqrt{9 y^{2} + 15 y + 18}}^{2} = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

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 $\sqrt{9 y^{2} + 15 y + 18}$ as ${(9 y^{2} + 15 y + 18)}^{\frac{1}{2}}$ .

${({(9 y^{2} + 15 y + 18)}^{\frac{1}{2}})}^{2} = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(9 y^{2} + 15 y + 18)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(9 y^{2} + 15 y + 18)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(9 y^{2} + 15 y + 18)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(9 y^{2} + 15 y + 18)}^{\frac{1}{2} \cdot 2} = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(9 y^{2} + 15 y + 18)}^{1} = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

Step 2.2.1.2

Simplify.

$9 y^{2} + 15 y + 18 = {\sqrt{5 y^{2} - 5 y - 7}}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Rewrite ${\sqrt{5 y^{2} - 5 y - 7}}^{2}$ as $5 y^{2} - 5 y - 7$ .

Tap for more steps...

Step 2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 y^{2} - 5 y - 7}$ as ${(5 y^{2} - 5 y - 7)}^{\frac{1}{2}}$ .

$9 y^{2} + 15 y + 18 = {({(5 y^{2} - 5 y - 7)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2

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

$9 y^{2} + 15 y + 18 = {(5 y^{2} - 5 y - 7)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3

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

$9 y^{2} + 15 y + 18 = {(5 y^{2} - 5 y - 7)}^{\frac{2}{2}}$

Step 2.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.1.4.1

Cancel the common factor.

$9 y^{2} + 15 y + 18 = {(5 y^{2} - 5 y - 7)}^{\frac{2}{2}}$

Step 2.3.1.4.2

Rewrite the expression.

$9 y^{2} + 15 y + 18 = {(5 y^{2} - 5 y - 7)}^{1}$

Step 2.3.1.5

Simplify.

$9 y^{2} + 15 y + 18 = 5 y^{2} - 5 y - 7$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $5 y^{2}$ from both sides of the equation.

$9 y^{2} + 15 y + 18 - 5 y^{2} = - 5 y - 7$

Step 3.1.2

Add $5 y$ to both sides of the equation.

$9 y^{2} + 15 y + 18 - 5 y^{2} + 5 y = - 7$

Step 3.1.3

Subtract $5 y^{2}$ from $9 y^{2}$ .

$4 y^{2} + 15 y + 18 + 5 y = - 7$

Step 3.1.4

Add $15 y$ and $5 y$ .

$4 y^{2} + 20 y + 18 = - 7$

Step 3.2

Add $7$ to both sides of the equation.

$4 y^{2} + 20 y + 18 + 7 = 0$

Step 3.3

Add $18$ and $7$ .

$4 y^{2} + 20 y + 25 = 0$

Step 3.4

Factor using the perfect square rule.

Tap for more steps...

Step 3.4.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

${(2 y)}^{2} + 20 y + 25 = 0$

Step 3.4.2

Rewrite $25$ as $5^{2}$ .

${(2 y)}^{2} + 20 y + 5^{2} = 0$

Step 3.4.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$20 y = 2 \cdot (2 y) \cdot 5$

Step 3.4.4

Rewrite the polynomial.

${(2 y)}^{2} + 2 \cdot (2 y) \cdot 5 + 5^{2} = 0$

Step 3.4.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 y$ and $b = 5$ .

${(2 y + 5)}^{2} = 0$

Step 3.5

Set the $2 y + 5$ equal to $0$ .

$2 y + 5 = 0$

Step 3.6

Solve for $y$ .

Tap for more steps...

Step 3.6.1

Subtract $5$ from both sides of the equation.

$2 y = - 5$

Step 3.6.2

Divide each term in $2 y = - 5$ by $2$ and simplify.

Tap for more steps...

Step 3.6.2.1

Divide each term in $2 y = - 5$ by $2$ .

$\frac{2 y}{2} = \frac{- 5}{2}$

Step 3.6.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{- 5}{2}$

Step 3.6.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 5}{2}$

Step 3.6.2.3

Simplify the right side.

Tap for more steps...

Step 3.6.2.3.1

Move the negative in front of the fraction.

$y = - \frac{5}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{5}{2}$

Decimal Form:

$y = - 2.5$

Mixed Number Form:

$y = - 2 \frac{1}{2}$