Algebra Examples

4 x^{2} + 5 x + 2 = 2 x^{2} + 7 x - 1

$4 x^{2} + 5 x + 2 = 2 x^{2} + 7 x - 1$

Step 1

Move all terms containing

x

$x$ to the left side of the equation.

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Step 1.1

Subtract

2 x^{2}

$2 x^{2}$ from both sides of the equation.

4 x^{2} + 5 x + 2 - 2 x^{2} = 7 x - 1

$4 x^{2} + 5 x + 2 - 2 x^{2} = 7 x - 1$

Step 1.2

Subtract

7 x

$7 x$ from both sides of the equation.

4 x^{2} + 5 x + 2 - 2 x^{2} - 7 x = - 1

$4 x^{2} + 5 x + 2 - 2 x^{2} - 7 x = - 1$

Step 1.3

Subtract

2 x^{2}

$2 x^{2}$ from

4 x^{2}

$4 x^{2}$ .

2 x^{2} + 5 x + 2 - 7 x = - 1

$2 x^{2} + 5 x + 2 - 7 x = - 1$

Step 1.4

Subtract

7 x

$7 x$ from

5 x

$5 x$ .

2 x^{2} - 2 x + 2 = - 1

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

2 x^{2} - 2 x + 2 = - 1

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

Step 2

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

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Step 2.1

Add

1

$1$ to both sides of the equation.

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

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

Step 2.2

Add

2

$2$ and

1

$1$ .

2 x^{2} - 2 x + 3 = 0

$2 x^{2} - 2 x + 3 = 0$

2 x^{2} - 2 x + 3 = 0

$2 x^{2} - 2 x + 3 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

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

Step 4

Substitute the values

a = 2

$a = 2$ ,

b = - 2

$b = - 2$ , and

c = 3

$c = 3$ into the quadratic formula and solve for

x

$x$ .

\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Step 5.1.2

Multiply

- 4 \cdot 2 \cdot 3

$- 4 \cdot 2 \cdot 3$ .

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Step 5.1.2.1

Multiply

- 4

$- 4$ by

2

$2$ .

x = \frac{2 \pm \sqrt{4 - 8 \cdot 3}}{2 \cdot 2}

$x = \frac{2 \pm \sqrt{4 - 8 \cdot 3}}{2 \cdot 2}$

Step 5.1.2.2

Multiply

- 8

$- 8$ by

3

$3$ .

x = \frac{2 \pm \sqrt{4 - 24}}{2 \cdot 2}

$x = \frac{2 \pm \sqrt{4 - 24}}{2 \cdot 2}$

x = \frac{2 \pm \sqrt{4 - 24}}{2 \cdot 2}

$x = \frac{2 \pm \sqrt{4 - 24}}{2 \cdot 2}$

Step 5.1.3

Subtract

24

$24$ from

4

$4$ .

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

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

Step 5.1.4

Rewrite

- 20

$- 20$ as

- 1 (20)

$- 1 (20)$ .

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

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

Step 5.1.5

Rewrite

\sqrt{- 1 (20)}

$\sqrt{- 1 (20)}$ as

\sqrt{- 1} \cdot \sqrt{20}

$\sqrt{- 1} \cdot \sqrt{20}$ .

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

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

Step 5.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x = \frac{2 \pm i \cdot \sqrt{20}}{2 \cdot 2}

$x = \frac{2 \pm i \cdot \sqrt{20}}{2 \cdot 2}$

Step 5.1.7

Rewrite

20

$20$ as

2^{2} \cdot 5

$2^{2} \cdot 5$ .

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Step 5.1.7.1

Factor

4

$4$ out of

20

$20$ .

x = \frac{2 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 2}

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

Step 5.1.7.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 2}

$x = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{2 \pm i \cdot (2 \sqrt{5})}{2 \cdot 2}$

Step 5.1.9

Move

$2$ to the left of

$i$ .

$x = \frac{2 \pm 2 i \sqrt{5}}{2 \cdot 2}$

Step 5.2

Multiply

$2$ by

$2$ .

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

Step 5.3

Simplify

$\frac{2 \pm 2 i \sqrt{5}}{4}$ .

$x = \frac{1 \pm i \sqrt{5}}{2}$

Step 6