Algebra Examples

$2 x^{2} - x + 55 = x + 5$

Step 1

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

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Step 1.1

Subtract $x$ from both sides of the equation.

$2 x^{2} - x + 55 - x = 5$

Step 1.2

Subtract $x$ from $- x$ .

$2 x^{2} - 2 x + 55 = 5$

Step 2

Subtract $5$ from both sides of the equation.

$2 x^{2} - 2 x + 55 - 5 = 0$

Step 3

Subtract $5$ from $55$ .

$2 x^{2} - 2 x + 50 = 0$

Step 4

Factor $2$ out of $2 x^{2} - 2 x + 50$ .

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Step 4.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 2 x + 50 = 0$

Step 4.2

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

$2 (x^{2}) + 2 (- x) + 50 = 0$

Step 4.3

Factor $2$ out of $50$ .

$2 (x^{2}) + 2 (- x) + 2 (25) = 0$

Step 4.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

$2 (x^{2} - x) + 2 (25) = 0$

Step 4.5

Factor $2$ out of $2 (x^{2} - x) + 2 (25)$ .

$2 (x^{2} - x + 25) = 0$

Step 5

Divide each term in $2 (x^{2} - x + 25) = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 (x^{2} - x + 25) = 0$ by $2$ .

$\frac{2 (x^{2} - x + 25)}{2} = \frac{0}{2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - x + 25)}{2} = \frac{0}{2}$

Step 5.2.1.2

Divide $x^{2} - x + 25$ by $1$ .

$x^{2} - x + 25 = \frac{0}{2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$x^{2} - x + 25 = 0$

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = - 1$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 25)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 25}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot 25$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 25}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $25$ .

$x = \frac{1 \pm \sqrt{1 - 100}}{2 \cdot 1}$

Step 8.1.3

Subtract $100$ from $1$ .

$x = \frac{1 \pm \sqrt{- 99}}{2 \cdot 1}$

Step 8.1.4

Rewrite $- 99$ as $- 1 (99)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 99}}{2 \cdot 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (99)}$ as $\sqrt{- 1} \cdot \sqrt{99}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{99}}{2 \cdot 1}$

Step 8.1.6

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

$x = \frac{1 \pm i \cdot \sqrt{99}}{2 \cdot 1}$

Step 8.1.7

Rewrite $99$ as $3^{2} \cdot 11$ .

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Step 8.1.7.1

Factor $9$ out of $99$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (11)}}{2 \cdot 1}$

Step 8.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{1 \pm i \cdot \sqrt{3^{2} \cdot 11}}{2 \cdot 1}$

Step 8.1.8

Pull terms out from under the radical.

$x = \frac{1 \pm i \cdot (3 \sqrt{11})}{2 \cdot 1}$

Step 8.1.9

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

$x = \frac{1 \pm 3 i \sqrt{11}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm 3 i \sqrt{11}}{2}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{1 + 3 i \sqrt{11}}{2}, \frac{1 - 3 i \sqrt{11}}{2}$