Algebra Examples

$\sqrt{z^{2} + 6 z + 3} = 3 z + 1$

Step 1

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

${\sqrt{z^{2} + 6 z + 3}}^{2} = {(3 z + 1)}^{2}$

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 $\sqrt{z^{2} + 6 z + 3}$ as ${(z^{2} + 6 z + 3)}^{\frac{1}{2}}$ .

${({(z^{2} + 6 z + 3)}^{\frac{1}{2}})}^{2} = {(3 z + 1)}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(z^{2} + 6 z + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(z^{2} + 6 z + 3)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(z^{2} + 6 z + 3)}^{\frac{1}{2} \cdot 2} = {(3 z + 1)}^{2}$

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.

${(z^{2} + 6 z + 3)}^{\frac{1}{2} \cdot 2} = {(3 z + 1)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(z^{2} + 6 z + 3)}^{1} = {(3 z + 1)}^{2}$

Step 2.2.1.2

Simplify.

$z^{2} + 6 z + 3 = {(3 z + 1)}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify ${(3 z + 1)}^{2}$ .

Tap for more steps...

Step 2.3.1.1

Rewrite ${(3 z + 1)}^{2}$ as $(3 z + 1) (3 z + 1)$ .

$z^{2} + 6 z + 3 = (3 z + 1) (3 z + 1)$

Step 2.3.1.2

Expand $(3 z + 1) (3 z + 1)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.2.1

Apply the distributive property.

$z^{2} + 6 z + 3 = 3 z (3 z + 1) + 1 (3 z + 1)$

Step 2.3.1.2.2

Apply the distributive property.

$z^{2} + 6 z + 3 = 3 z (3 z) + 3 z \cdot 1 + 1 (3 z + 1)$

Step 2.3.1.2.3

Apply the distributive property.

$z^{2} + 6 z + 3 = 3 z (3 z) + 3 z \cdot 1 + 1 (3 z) + 1 \cdot 1$

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.

$z^{2} + 6 z + 3 = 3 \cdot 3 z \cdot z + 3 z \cdot 1 + 1 (3 z) + 1 \cdot 1$

Step 2.3.1.3.1.2

Multiply $z$ by $z$ by adding the exponents.

Tap for more steps...

Step 2.3.1.3.1.2.1

Move $z$ .

$z^{2} + 6 z + 3 = 3 \cdot 3 (z \cdot z) + 3 z \cdot 1 + 1 (3 z) + 1 \cdot 1$

Step 2.3.1.3.1.2.2

Multiply $z$ by $z$ .

$z^{2} + 6 z + 3 = 3 \cdot 3 z^{2} + 3 z \cdot 1 + 1 (3 z) + 1 \cdot 1$

Step 2.3.1.3.1.3

Multiply $3$ by $3$ .

$z^{2} + 6 z + 3 = 9 z^{2} + 3 z \cdot 1 + 1 (3 z) + 1 \cdot 1$

Step 2.3.1.3.1.4

Multiply $3$ by $1$ .

$z^{2} + 6 z + 3 = 9 z^{2} + 3 z + 1 (3 z) + 1 \cdot 1$

Step 2.3.1.3.1.5

Multiply $3 z$ by $1$ .

$z^{2} + 6 z + 3 = 9 z^{2} + 3 z + 3 z + 1 \cdot 1$

Step 2.3.1.3.1.6

Multiply $1$ by $1$ .

$z^{2} + 6 z + 3 = 9 z^{2} + 3 z + 3 z + 1$

Step 2.3.1.3.2

Add $3 z$ and $3 z$ .

$z^{2} + 6 z + 3 = 9 z^{2} + 6 z + 1$

Step 3

Solve for $z$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $9 z^{2}$ from both sides of the equation.

$z^{2} + 6 z + 3 - 9 z^{2} = 6 z + 1$

Step 3.1.2

Subtract $6 z$ from both sides of the equation.

$z^{2} + 6 z + 3 - 9 z^{2} - 6 z = 1$

Step 3.1.3

Combine the opposite terms in $z^{2} + 6 z + 3 - 9 z^{2} - 6 z$ .

Tap for more steps...

Step 3.1.3.1

Subtract $6 z$ from $6 z$ .

$z^{2} + 3 - 9 z^{2} + 0 = 1$

Step 3.1.3.2

Add $z^{2} + 3 - 9 z^{2}$ and $0$ .

$z^{2} + 3 - 9 z^{2} = 1$

Step 3.1.4

Subtract $9 z^{2}$ from $z^{2}$ .

$- 8 z^{2} + 3 = 1$

Step 3.2

Move all terms not containing $z$ to the right side of the equation.

Tap for more steps...

Step 3.2.1

Subtract $3$ from both sides of the equation.

$- 8 z^{2} = 1 - 3$

Step 3.2.2

Subtract $3$ from $1$ .

$- 8 z^{2} = - 2$

Step 3.3

Divide each term in $- 8 z^{2} = - 2$ by $- 8$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $- 8 z^{2} = - 2$ by $- 8$ .

$\frac{- 8 z^{2}}{- 8} = \frac{- 2}{- 8}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $- 8$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 8 z^{2}}{- 8} = \frac{- 2}{- 8}$

Step 3.3.2.1.2

Divide $z^{2}$ by $1$ .

$z^{2} = \frac{- 2}{- 8}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Cancel the common factor of $- 2$ and $- 8$ .

Tap for more steps...

Step 3.3.3.1.1

Factor $- 2$ out of $- 2$ .

$z^{2} = \frac{- 2 (1)}{- 8}$

Step 3.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.3.3.1.2.1

Factor $- 2$ out of $- 8$ .

$z^{2} = \frac{- 2 \cdot 1}{- 2 \cdot 4}$

Step 3.3.3.1.2.2

Cancel the common factor.

$z^{2} = \frac{- 2 \cdot 1}{- 2 \cdot 4}$

Step 3.3.3.1.2.3

Rewrite the expression.

$z^{2} = \frac{1}{4}$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{\frac{1}{4}}$

Step 3.5

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Tap for more steps...

Step 3.5.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$z = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 3.5.2

Any root of $1$ is $1$ .

$z = \pm \frac{1}{\sqrt{4}}$

Step 3.5.3

Simplify the denominator.

Tap for more steps...

Step 3.5.3.1

Rewrite $4$ as $2^{2}$ .

$z = \pm \frac{1}{\sqrt{2^{2}}}$

Step 3.5.3.2

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm \frac{1}{2}$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$z = \frac{1}{2}$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$z = \frac{1}{2}, - \frac{1}{2}$

Step 4

Exclude the solutions that do not make $\sqrt{z^{2} + 6 z + 3} = 3 z + 1$ true.

$z = \frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$z = \frac{1}{2}$

Decimal Form:

$z = 0.5$