Pre-Algebra Examples

$(4 x + 1) (x - 3) = 9$

Step 1

Simplify $(4 x + 1) (x - 3)$ .

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Step 1.1

Expand $(4 x + 1) (x - 3)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$4 x (x - 3) + 1 (x - 3) = 9$

Step 1.1.2

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 3 + 1 (x - 3) = 9$

Step 1.1.3

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 3 + 1 x + 1 \cdot - 3 = 9$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

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Step 1.2.1.1.1

Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 3 + 1 x + 1 \cdot - 3 = 9$

Step 1.2.1.1.2

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot - 3 + 1 x + 1 \cdot - 3 = 9$

Step 1.2.1.2

Multiply $- 3$ by $4$ .

$4 x^{2} - 12 x + 1 x + 1 \cdot - 3 = 9$

Step 1.2.1.3

Multiply $x$ by $1$ .

$4 x^{2} - 12 x + x + 1 \cdot - 3 = 9$

Step 1.2.1.4

Multiply $- 3$ by $1$ .

$4 x^{2} - 12 x + x - 3 = 9$

Step 1.2.2

Add $- 12 x$ and $x$ .

$4 x^{2} - 11 x - 3 = 9$

Step 2

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

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Step 2.1

Subtract $9$ from both sides of the equation.

$4 x^{2} - 11 x - 3 - 9 = 0$

Step 2.2

Subtract $9$ from $- 3$ .

$4 x^{2} - 11 x - 12 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = - 11$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 12$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{11 \pm \sqrt{121 - 16 \cdot - 12}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{11 \pm \sqrt{121 + 192}}{2 \cdot 4}$

Step 5.1.3

Add $121$ and $192$ .

$x = \frac{11 \pm \sqrt{313}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{11 \pm \sqrt{313}}{8}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 12$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{11 \pm \sqrt{121 - 16 \cdot - 12}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{11 \pm \sqrt{121 + 192}}{2 \cdot 4}$

Step 6.1.3

Add $121$ and $192$ .

$x = \frac{11 \pm \sqrt{313}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{11 \pm \sqrt{313}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{11 + \sqrt{313}}{8}$

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 $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 12$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{11 \pm \sqrt{121 - 16 \cdot - 12}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{11 \pm \sqrt{121 + 192}}{2 \cdot 4}$

Step 7.1.3

Add $121$ and $192$ .

$x = \frac{11 \pm \sqrt{313}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{11 \pm \sqrt{313}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{11 - \sqrt{313}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{11 + \sqrt{313}}{8}, \frac{11 - \sqrt{313}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 + \sqrt{313}}{8}, \frac{11 - \sqrt{313}}{8}$

Decimal Form:

$x = 3.58647575 \dots, - 0.83647575 \dots$