Pre-Algebra Examples

$2 x (2 x - 3) = 6 + x$

Step 1

Simplify $2 x (2 x - 3)$ .

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Step 1.1

Rewrite.

$0 + 0 + 2 x (2 x - 3) = 6 + x$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 3 = 6 + x$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 3 = 6 + x$

Step 1.2.2.2

Multiply $- 3$ by $2$ .

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

Step 1.3

Simplify each term.

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Step 1.3.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.1

Move $x$ .

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

Step 1.3.1.2

Multiply $x$ by $x$ .

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

Step 1.3.2

Multiply $2$ by $2$ .

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

Step 2

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

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Step 2.1

Subtract $x$ from both sides of the equation.

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

Step 2.2

Subtract $x$ from $- 6 x$ .

$4 x^{2} - 7 x = 6$

Step 3

Subtract $6$ from both sides of the equation.

$4 x^{2} - 7 x - 6 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = - 7$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (4 \cdot - 6)}}{2 \cdot 4}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 6}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 6$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 6}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 6$ .

$x = \frac{7 \pm \sqrt{49 + 96}}{2 \cdot 4}$

Step 6.1.3

Add $49$ and $96$ .

$x = \frac{7 \pm \sqrt{145}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{145}}{8}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 6}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 6$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 6}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 6$ .

$x = \frac{7 \pm \sqrt{49 + 96}}{2 \cdot 4}$

Step 7.1.3

Add $49$ and $96$ .

$x = \frac{7 \pm \sqrt{145}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{145}}{8}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{145}}{8}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 4 \cdot - 6}}{2 \cdot 4}$

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot - 6$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{7 \pm \sqrt{49 - 16 \cdot - 6}}{2 \cdot 4}$

Step 8.1.2.2

Multiply $- 16$ by $- 6$ .

$x = \frac{7 \pm \sqrt{49 + 96}}{2 \cdot 4}$

Step 8.1.3

Add $49$ and $96$ .

$x = \frac{7 \pm \sqrt{145}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$x = \frac{7 \pm \sqrt{145}}{8}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{145}}{8}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{145}}{8}, \frac{7 - \sqrt{145}}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 + \sqrt{145}}{8}, \frac{7 - \sqrt{145}}{8}$

Decimal Form:

$x = 2.38019932 \dots, - 0.63019932 \dots$