Algebra Examples

x (x - 2) = 48

$x (x - 2) = 48$

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify

x (x - 2)

$x (x - 2)$ .

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Step 1.1.1.1

Apply the distributive property.

x \cdot x + x \cdot - 2 = 48

$x \cdot x + x \cdot - 2 = 48$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot - 2 = 48

$x^{2} + x \cdot - 2 = 48$

Step 1.1.1.2.2

Move

- 2

$- 2$ to the left of

x

$x$ .

x^{2} - 2 x = 48

$x^{2} - 2 x = 48$

x^{2} - 2 x = 48

$x^{2} - 2 x = 48$

x^{2} - 2 x = 48

$x^{2} - 2 x = 48$

x^{2} - 2 x = 48

$x^{2} - 2 x = 48$

Step 1.2

Subtract

48

$48$ from both sides of the equation.

x^{2} - 2 x - 48 = 0

$x^{2} - 2 x - 48 = 0$

x^{2} - 2 x - 48 = 0

$x^{2} - 2 x - 48 = 0$

Step 2

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 3

Substitute the values

a = 1

$a = 1$ ,

b = - 2

$b = - 2$ , and

c = - 48

$c = - 48$ into the quadratic formula and solve for

x

$x$ .

\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 48)}}{2 \cdot 1}

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot - 48)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 48}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 48}}{2 \cdot 1}$

Step 4.1.2

Multiply

- 4 \cdot 1 \cdot - 48

$- 4 \cdot 1 \cdot - 48$ .

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Step 4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

x = \frac{2 \pm \sqrt{4 - 4 \cdot - 48}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 48}}{2 \cdot 1}$

Step 4.1.2.2

Multiply

- 4

$- 4$ by

- 48

$- 48$ .

x = \frac{2 \pm \sqrt{4 + 192}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{4 + 192}}{2 \cdot 1}$

x = \frac{2 \pm \sqrt{4 + 192}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{4 + 192}}{2 \cdot 1}$

Step 4.1.3

Add

4

$4$ and

192

$192$ .

x = \frac{2 \pm \sqrt{196}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{196}}{2 \cdot 1}$

Step 4.1.4

Rewrite

196

$196$ as

14^{2}

$14^{2}$ .

x = \frac{2 \pm \sqrt{14^{2}}}{2 \cdot 1}

$x = \frac{2 \pm \sqrt{14^{2}}}{2 \cdot 1}$

Step 4.1.5

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

$x = \frac{2 \pm 14}{2 \cdot 1}$

Step 4.2

Multiply

$2$ by

$1$ .

$x = \frac{2 \pm 14}{2}$

Step 4.3

Simplify

$\frac{2 \pm 14}{2}$ .

$x = 1 \pm 7$

Step 5

The final answer is the combination of both solutions.

$x = 8, - 6$