Algebra Examples

$\sqrt{25 y^{2} + 40 y + 16} = 5 y + 4$

Step 1

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

${\sqrt{25 y^{2} + 40 y + 16}}^{2} = {(5 y + 4)}^{2}$

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 $\sqrt{25 y^{2} + 40 y + 16}$ as ${(25 y^{2} + 40 y + 16)}^{\frac{1}{2}}$ .

${({(25 y^{2} + 40 y + 16)}^{\frac{1}{2}})}^{2} = {(5 y + 4)}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(25 y^{2} + 40 y + 16)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(25 y^{2} + 40 y + 16)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(25 y^{2} + 40 y + 16)}^{\frac{1}{2} \cdot 2} = {(5 y + 4)}^{2}$

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.

${(25 y^{2} + 40 y + 16)}^{\frac{1}{2} \cdot 2} = {(5 y + 4)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(25 y^{2} + 40 y + 16)}^{1} = {(5 y + 4)}^{2}$

Step 2.2.1.2

Simplify.

$25 y^{2} + 40 y + 16 = {(5 y + 4)}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify ${(5 y + 4)}^{2}$ .

Tap for more steps...

Step 2.3.1.1

Rewrite ${(5 y + 4)}^{2}$ as $(5 y + 4) (5 y + 4)$ .

$25 y^{2} + 40 y + 16 = (5 y + 4) (5 y + 4)$

Step 2.3.1.2

Expand $(5 y + 4) (5 y + 4)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.2.1

Apply the distributive property.

$25 y^{2} + 40 y + 16 = 5 y (5 y + 4) + 4 (5 y + 4)$

Step 2.3.1.2.2

Apply the distributive property.

$25 y^{2} + 40 y + 16 = 5 y (5 y) + 5 y \cdot 4 + 4 (5 y + 4)$

Step 2.3.1.2.3

Apply the distributive property.

$25 y^{2} + 40 y + 16 = 5 y (5 y) + 5 y \cdot 4 + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.3.1.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$25 y^{2} + 40 y + 16 = 5 \cdot 5 y \cdot y + 5 y \cdot 4 + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 2.3.1.3.1.2.1

Move $y$ .

$25 y^{2} + 40 y + 16 = 5 \cdot 5 (y \cdot y) + 5 y \cdot 4 + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3.1.2.2

Multiply $y$ by $y$ .

$25 y^{2} + 40 y + 16 = 5 \cdot 5 y^{2} + 5 y \cdot 4 + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3.1.3

Multiply $5$ by $5$ .

$25 y^{2} + 40 y + 16 = 25 y^{2} + 5 y \cdot 4 + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3.1.4

Multiply $4$ by $5$ .

$25 y^{2} + 40 y + 16 = 25 y^{2} + 20 y + 4 (5 y) + 4 \cdot 4$

Step 2.3.1.3.1.5

Multiply $5$ by $4$ .

$25 y^{2} + 40 y + 16 = 25 y^{2} + 20 y + 20 y + 4 \cdot 4$

Step 2.3.1.3.1.6

Multiply $4$ by $4$ .

$25 y^{2} + 40 y + 16 = 25 y^{2} + 20 y + 20 y + 16$

Step 2.3.1.3.2

Add $20 y$ and $20 y$ .

$25 y^{2} + 40 y + 16 = 25 y^{2} + 40 y + 16$

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 $25 y^{2}$ from both sides of the equation.

$25 y^{2} + 40 y + 16 - 25 y^{2} = 40 y + 16$

Step 3.1.2

Subtract $40 y$ from both sides of the equation.

$25 y^{2} + 40 y + 16 - 25 y^{2} - 40 y = 16$

Step 3.1.3

Combine the opposite terms in $25 y^{2} + 40 y + 16 - 25 y^{2} - 40 y$ .

Tap for more steps...

Step 3.1.3.1

Subtract $25 y^{2}$ from $25 y^{2}$ .

$40 y + 16 + 0 - 40 y = 16$

Step 3.1.3.2

Add $40 y + 16$ and $0$ .

$40 y + 16 - 40 y = 16$

Step 3.1.3.3

Subtract $40 y$ from $40 y$ .

$0 + 16 = 16$

Step 3.1.3.4

Add $0$ and $16$ .

$16 = 16$

Step 3.2

Since $16 = 16$ , the equation will always be true for any value of $y$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$