Algebra Examples

$\frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2}$

Step 1

Set $\frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2}$ equal to $0$ .

$\frac{1}{2} x^{2} + \frac{5}{2} x - \frac{3}{2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} + \frac{5}{2} x - \frac{3}{2} = 0$

Step 2.1.2

Combine $\frac{5}{2}$ and $x$ .

$\frac{x^{2}}{2} + \frac{5 x}{2} - \frac{3}{2} = 0$

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Step 2.2.2.1.2

Rewrite the expression.

$x^{2} + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Step 2.2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.2.1

Cancel the common factor.

$x^{2} + 2 (\frac{5 x}{2}) + 2 (- \frac{3}{2}) = 0$

Step 2.2.2.2.2

Rewrite the expression.

$x^{2} + 5 x + 2 (- \frac{3}{2}) = 0$

Step 2.2.2.3

Cancel the common factor of $2$ .

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Step 2.2.2.3.1

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

$x^{2} + 5 x + 2 (\frac{- 3}{2}) = 0$

Step 2.2.2.3.2

Cancel the common factor.

$x^{2} + 5 x + 2 (\frac{- 3}{2}) = 0$

Step 2.2.2.3.3

Rewrite the expression.

$x^{2} + 5 x - 3 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 1$ , $b = 5$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 3)}}{2 \cdot 1}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 3}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 3$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 3}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $- 3$ .

$x = \frac{- 5 \pm \sqrt{25 + 12}}{2 \cdot 1}$

Step 2.5.1.3

Add $25$ and $12$ .

$x = \frac{- 5 \pm \sqrt{37}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm \sqrt{37}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{5 - \sqrt{37}}{2}, - \frac{5 + \sqrt{37}}{2}$

$x = \frac{- 5 \pm \sqrt{37}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm \sqrt{37}}{2}$

Decimal Form:

$x = 0.54138126 \dots, - 5.54138126 \dots$

Step 4