Algebra Examples

$\frac{1}{2} x^{2} - \frac{5}{4} x - 3 = 0$

Step 1

Find $a$ , $b$ , and $c$ .

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

$\frac{x^{2}}{2} - \frac{5}{4} x - 3 = 0$

Step 1.1.1.2

Combine $x$ and $\frac{5}{4}$ .

$\frac{x^{2}}{2} - \frac{x \cdot 5}{4} - 3 = 0$

Step 1.1.1.3

Move $5$ to the left of $x$ .

$\frac{x^{2}}{2} - \frac{5 x}{4} - 3 = 0$

Step 1.2

Once the quadratic is in standard form, the values of $a$ , $b$ , and $c$ can be found.

$a x^{2} + b x + c$

Step 1.3

Use the standard form of the equation to find $a$ , $b$ , and $c$ for this quadratic.

$a = \frac{1}{2}$ , $b = - \frac{5}{4}$ , $c = - 3$

Step 2

Substitute the values into the sum of roots formula $- \frac{b}{a}$ .

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{5}{4} \cdot 2$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $4$ .

$\frac{5}{2 (2)} \cdot 2$

Step 2.2.2

Cancel the common factor.

$\frac{5}{2 \cdot 2} \cdot 2$

Step 2.2.3

Rewrite the expression.

$\frac{5}{2}$

Step 3

Substitute the values into the product of roots formula $\frac{c}{a}$ .

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$- 3 \cdot 2$

Step 3.2

Multiply $- 3$ by $2$ .

$- 6$

Step 4

List the results.

Sum of the roots is $\frac{5}{2}$

Product of the roots is $- 6$