Algebra Examples

$\sqrt{2 x + 6 + 4} = x + 3$

Step 1

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

${\sqrt{2 x + 6 + 4}}^{2} = {(x + 3)}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 6 + 4}$ as ${(2 x + 6 + 4)}^{\frac{1}{2}}$ .

${({(2 x + 6 + 4)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(2 x + 6 + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(2 x + 6 + 4)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(2 x + 6 + 4)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(2 x + 6 + 4)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(2 x + 6 + 4)}^{1} = {(x + 3)}^{2}$

Step 2.2.1.2

Add $6$ and $4$ .

${(2 x + 10)}^{1} = {(x + 3)}^{2}$

Step 2.2.1.3

Simplify.

$2 x + 10 = {(x + 3)}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(x + 3)}^{2}$ .

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Step 2.3.1.1

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

$2 x + 10 = (x + 3) (x + 3)$

Step 2.3.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.3.1.2.1

Apply the distributive property.

$2 x + 10 = x (x + 3) + 3 (x + 3)$

Step 2.3.1.2.2

Apply the distributive property.

$2 x + 10 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 2.3.1.2.3

Apply the distributive property.

$2 x + 10 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3

Simplify and combine like terms.

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Step 2.3.1.3.1

Simplify each term.

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Step 2.3.1.3.1.1

Multiply $x$ by $x$ .

$2 x + 10 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.2

Move $3$ to the left of $x$ .

$2 x + 10 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 2.3.1.3.1.3

Multiply $3$ by $3$ .

$2 x + 10 = x^{2} + 3 x + 3 x + 9$

Step 2.3.1.3.2

Add $3 x$ and $3 x$ .

$2 x + 10 = x^{2} + 6 x + 9$

Step 3

Solve for $x$ .

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Step 3.1

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

$x^{2} + 6 x + 9 = 2 x + 10$

Step 3.2

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

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Step 3.2.1

Subtract $2 x$ from both sides of the equation.

$x^{2} + 6 x + 9 - 2 x = 10$

Step 3.2.2

Subtract $2 x$ from $6 x$ .

$x^{2} + 4 x + 9 = 10$

Step 3.3

Move all terms to the left side of the equation and simplify.

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Step 3.3.1

Subtract $10$ from both sides of the equation.

$x^{2} + 4 x + 9 - 10 = 0$

Step 3.3.2

Subtract $10$ from $9$ .

$x^{2} + 4 x - 1 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 1$ , $b = 4$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 3.6.1.2

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

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Step 3.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1}}{2 \cdot 1}$

Step 3.6.1.2.2

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4}}{2 \cdot 1}$

Step 3.6.1.3

Add $16$ and $4$ .

$x = \frac{- 4 \pm \sqrt{20}}{2 \cdot 1}$

Step 3.6.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.6.1.4.1

Factor $4$ out of $20$ .

$x = \frac{- 4 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 3.6.1.4.2

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 3.6.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 3.6.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 \sqrt{5}}{2}$

Step 3.6.3

Simplify $\frac{- 4 \pm 2 \sqrt{5}}{2}$ .

$x = - 2 \pm \sqrt{5}$

Step 3.7

The final answer is the combination of both solutions.

$x = - 2 + \sqrt{5}, - 2 - \sqrt{5}$

Step 4

Exclude the solutions that do not make $\sqrt{2 x + 6 + 4} = x + 3$ true.

$x = - 2 + \sqrt{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 2 + \sqrt{5}$

Decimal Form:

$x = 0.23606797 \dots$