Algebra Examples

$x + 3 = \sqrt{2 x - 1 + 5}$

Step 1

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

$\sqrt{2 x - 1 + 5} = x + 3$

Step 2

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

${\sqrt{2 x - 1 + 5}}^{2} = {(x + 3)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

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

${({(2 x - 1 + 5)}^{\frac{1}{2}})}^{2} = {(x + 3)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x - 1 + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x - 1 + 5)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(2 x - 1 + 5)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(2 x - 1 + 5)}^{\frac{1}{2} \cdot 2} = {(x + 3)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x - 1 + 5)}^{1} = {(x + 3)}^{2}$

Step 3.2.1.2

Add $- 1$ and $5$ .

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

Step 3.2.1.3

Simplify.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

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

$2 x + 4 = (x + 3) (x + 3)$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Apply the distributive property.

$2 x + 4 = x (x + 3) + 3 (x + 3)$

Step 3.3.1.2.2

Apply the distributive property.

$2 x + 4 = x \cdot x + x \cdot 3 + 3 (x + 3)$

Step 3.3.1.2.3

Apply the distributive property.

$2 x + 4 = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $x$ by $x$ .

$2 x + 4 = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3$

Step 3.3.1.3.1.2

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

$2 x + 4 = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3$

Step 3.3.1.3.1.3

Multiply $3$ by $3$ .

$2 x + 4 = x^{2} + 3 x + 3 x + 9$

Step 3.3.1.3.2

Add $3 x$ and $3 x$ .

$2 x + 4 = x^{2} + 6 x + 9$

Step 4

Solve for $x$ .

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Step 4.1

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

$x^{2} + 6 x + 9 = 2 x + 4$

Step 4.2

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

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Step 4.2.1

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

$x^{2} + 6 x + 9 - 2 x = 4$

Step 4.2.2

Subtract $2 x$ from $6 x$ .

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

Step 4.3

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

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Step 4.3.1

Subtract $4$ from both sides of the equation.

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

Step 4.3.2

Subtract $4$ from $9$ .

$x^{2} + 4 x + 5 = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

Step 4.5

Substitute the values $a = 1$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 4.6.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 4.6.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 4.6.1.3

Subtract $20$ from $16$ .

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

Step 4.6.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.6.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 4.6.1.6

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

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

Step 4.6.1.7

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

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

Step 4.6.1.8

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

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

Step 4.6.1.9

Move $2$ to the left of $i$ .

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

Step 4.6.2

Multiply $2$ by $1$ .

$x = \frac{- 4 \pm 2 i}{2}$

Step 4.6.3

Simplify $\frac{- 4 \pm 2 i}{2}$ .

$x = - 2 \pm i$

Step 4.7

The final answer is the combination of both solutions.

$x = - 2 + i, - 2 - i$