Algebra Examples

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

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $2 (x - 3)$ .

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Step 1.1.1.1

Apply the distributive property.

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

Step 1.1.1.2

Multiply $2$ by $- 3$ .

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

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

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

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

Step 1.2.2

Add $6$ to both sides of the equation.

${(x + 1)}^{2} - 2 x + 6 = 0$

Step 1.3

Simplify ${(x + 1)}^{2} - 2 x + 6$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$(x + 1) (x + 1) - 2 x + 6 = 0$

Step 1.3.1.2

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.3.1.2.1

Apply the distributive property.

$x (x + 1) + 1 (x + 1) - 2 x + 6 = 0$

Step 1.3.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1) - 2 x + 6 = 0$

Step 1.3.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - 2 x + 6 = 0$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 - 2 x + 6 = 0$

Step 1.3.1.3.1.2

Multiply $x$ by $1$ .

$x^{2} + x + 1 x + 1 \cdot 1 - 2 x + 6 = 0$

Step 1.3.1.3.1.3

Multiply $x$ by $1$ .

$x^{2} + x + x + 1 \cdot 1 - 2 x + 6 = 0$

Step 1.3.1.3.1.4

Multiply $1$ by $1$ .

$x^{2} + x + x + 1 - 2 x + 6 = 0$

Step 1.3.1.3.2

Add $x$ and $x$ .

$x^{2} + 2 x + 1 - 2 x + 6 = 0$

Step 1.3.2

Combine the opposite terms in $x^{2} + 2 x + 1 - 2 x + 6$ .

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Step 1.3.2.1

Subtract $2 x$ from $2 x$ .

$x^{2} + 0 + 1 + 6 = 0$

Step 1.3.2.2

Add $x^{2}$ and $0$ .

$x^{2} + 1 + 6 = 0$

Step 1.3.3

Add $1$ and $6$ .

$x^{2} + 7 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 0$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 7}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{0 \pm \sqrt{0 - 28}}{2 \cdot 1}$

Step 4.1.3

Subtract $28$ from $0$ .

$x = \frac{0 \pm \sqrt{- 28}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 28$ as $- 1 (28)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 28}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (28)}$ as $\sqrt{- 1} \cdot \sqrt{28}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{28}}{2 \cdot 1}$

Step 4.1.6

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

$x = \frac{0 \pm i \cdot \sqrt{28}}{2 \cdot 1}$

Step 4.1.7

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.1.7.1

Factor $4$ out of $28$ .

$x = \frac{0 \pm i \cdot \sqrt{4 (7)}}{2 \cdot 1}$

Step 4.1.7.2

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

$x = \frac{0 \pm i \cdot \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (2 \sqrt{7})}{2 \cdot 1}$

Step 4.1.9

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

$x = \frac{0 \pm 2 i \sqrt{7}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{0 \pm 2 i \sqrt{7}}{2}$

Step 4.3

Simplify $\frac{0 \pm 2 i \sqrt{7}}{2}$ .

$x = \pm i \sqrt{7}$