Apply the Quadratic Formula (x-3)(x+11/4)

$(x - 3) (x + \frac{11}{4})$

Simplify $(x - 3) (x + \frac{11}{4})$ .

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Expand $(x - 3) (x + \frac{11}{4})$ using the FOIL Method.

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Apply the distributive property.

$x (x + \frac{11}{4}) - 3 (x + \frac{11}{4}) = 0$

Apply the distributive property.

$x \cdot x + x \frac{11}{4} - 3 (x + \frac{11}{4}) = 0$

Apply the distributive property.

$x \cdot x + x \frac{11}{4} - 3 x - 3 (\frac{11}{4}) = 0$

Simplify and combine like terms.

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Simplify each term.

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Multiply $x$ by $x$ .

$x^{2} + x \frac{11}{4} - 3 x - 3 (\frac{11}{4}) = 0$

Combine $x$ and $\frac{11}{4}$ .

$x^{2} + \frac{x \cdot 11}{4} - 3 x - 3 (\frac{11}{4}) = 0$

Move $11$ to the left of $x$ .

$x^{2} + \frac{11 \cdot x}{4} - 3 x - 3 (\frac{11}{4}) = 0$

Multiply $- 3 (\frac{11}{4})$ .

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Combine $- 3$ and $\frac{11}{4}$ .

$x^{2} + \frac{11 x}{4} - 3 x + \frac{- 3 \cdot 11}{4} = 0$

Multiply $- 3$ by $11$ .

$x^{2} + \frac{11 x}{4} - 3 x + \frac{- 33}{4} = 0$

Move the negative in front of the fraction.

$x^{2} + \frac{11 x}{4} - 3 x - \frac{33}{4} = 0$

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + \frac{11 x}{4} - 3 x \cdot \frac{4}{4} - \frac{33}{4} = 0$

Combine $- 3 x$ and $\frac{4}{4}$ .

$x^{2} + \frac{11 x}{4} + \frac{- 3 x \cdot 4}{4} - \frac{33}{4} = 0$

Combine the numerators over the common denominator.

$x^{2} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4} = 0$

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} \cdot \frac{4}{4} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4} = 0$

Combine $x^{2}$ and $\frac{4}{4}$ .

$\frac{x^{2} \cdot 4}{4} + \frac{11 x - 3 x \cdot 4}{4} - \frac{33}{4} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 + 11 x - 3 x \cdot 4}{4} - \frac{33}{4} = 0$

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 4 + 11 x - 3 x \cdot 4 - 33}{4} = 0$

Simplify the numerator.

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Move $4$ to the left of $x^{2}$ .

$\frac{4 \cdot x^{2} + 11 x - 3 x \cdot 4 - 33}{4} = 0$

Multiply $4$ by $- 3$ .

$\frac{4 x^{2} + 11 x - 12 x - 33}{4} = 0$

Subtract $12 x$ from $11 x$ .

$\frac{4 x^{2} - x - 33}{4} = 0$

Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 33 = - 132$ and whose sum is $b = - 1$ .

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Factor $- 1$ out of $- x$ .

$\frac{4 x^{2} - (x) - 33}{4} = 0$

Rewrite $- 1$ as $11$ plus $- 12$

$\frac{4 x^{2} + (11 - 12) x - 33}{4} = 0$

Apply the distributive property.

$\frac{4 x^{2} + 11 x - 12 x - 33}{4} = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{(4 x^{2} + 11 x) - 12 x - 33}{4} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (4 x + 11) - 3 (4 x + 11)}{4} = 0$

Factor the polynomial by factoring out the greatest common factor, $4 x + 11$ .

$\frac{(4 x + 11) (x - 3)}{4} = 0$

Multiply through by the least common denominator $4$ , then simplify.

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Cancel the common factor of $4$ .

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Cancel the common factor.

$4 (\frac{(4 x + 11) (x - 3)}{4}) = 0$

Rewrite the expression.

$(4 x + 11) (x - 3) = 0$

Expand $(4 x + 11) (x - 3)$ using the FOIL Method.

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Apply the distributive property.

$4 x (x - 3) + 11 (x - 3) = 0$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 3 + 11 (x - 3) = 0$

Apply the distributive property.

$4 x \cdot x + 4 x \cdot - 3 + 11 x + 11 \cdot - 3 = 0$

Simplify and combine like terms.

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Simplify each term.

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

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Move $x$ .

$4 (x \cdot x) + 4 x \cdot - 3 + 11 x + 11 \cdot - 3 = 0$

Multiply $x$ by $x$ .

$4 x^{2} + 4 x \cdot - 3 + 11 x + 11 \cdot - 3 = 0$

Multiply $- 3$ by $4$ .

$4 x^{2} - 12 x + 11 x + 11 \cdot - 3 = 0$

Multiply $11$ by $- 3$ .

$4 x^{2} - 12 x + 11 x - 33 = 0$

Add $- 12 x$ and $11 x$ .

$4 x^{2} - x - 33 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 4$ , $b = - 1$ , and $c = - 33$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (4 \cdot - 33)}}{2 \cdot 4}$

Simplify.

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Simplify the numerator.

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

$x = \frac{1 \pm \sqrt{1 - 4 \cdot (4 \cdot - 33)}}{2 \cdot 4}$

Multiply $4$ by $- 33$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 132}}{2 \cdot 4}$

Multiply $- 4$ by $- 132$ .

$x = \frac{1 \pm \sqrt{1 + 528}}{2 \cdot 4}$

Add $1$ and $528$ .

$x = \frac{1 \pm \sqrt{529}}{2 \cdot 4}$

Rewrite $529$ as $23^{2}$ .

$x = \frac{1 \pm \sqrt{23^{2}}}{2 \cdot 4}$

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

$x = \frac{1 \pm 23}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{1 \pm 23}{8}$

The final answer is the combination of both solutions.

$x = 3, - \frac{11}{4}$

Apply the Quadratic Formula (x-3)(x+11/4)