Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Combine $x$ and $\frac{11}{2}$ .

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

Step 1.2

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

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

Step 2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $2$ by $2$ .

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

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Move the leading negative in $- \frac{11 x}{2}$ into the numerator.

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 2.2.3.2

Cancel the common factor.

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

Step 2.2.3.3

Rewrite the expression.

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

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 $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (4 \cdot - 3)}}{2 \cdot 4}$

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 $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 3$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{11 \pm \sqrt{121 - 16 \cdot - 3}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{11 \pm \sqrt{121 + 48}}{2 \cdot 4}$

Step 5.1.3

Add $121$ and $48$ .

$x = \frac{11 \pm \sqrt{169}}{2 \cdot 4}$

Step 5.1.4

Rewrite $169$ as $13^{2}$ .

$x = \frac{11 \pm \sqrt{13^{2}}}{2 \cdot 4}$

Step 5.1.5

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

$x = \frac{11 \pm 13}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{11 \pm 13}{8}$

Step 6

The final answer is the combination of both solutions.

$x = 3, - \frac{1}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 3, - \frac{1}{4}$

Decimal Form:

$x = 3, - 0.25$