Finite Math Examples

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

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

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

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

Step 1.2

Subtract $1$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (5 x) + 2 \cdot - 1 = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (5 x) + 2 \cdot - 1 = 0$

Step 2.2.1.2

Rewrite the expression.

$x^{2} + 2 (5 x) + 2 \cdot - 1 = 0$

Step 2.2.2

Multiply $5$ by $2$ .

$x^{2} + 10 x + 2 \cdot - 1 = 0$

Step 2.2.3

Multiply $2$ by $- 1$ .

$x^{2} + 10 x - 2 = 0$

Step 3

The discriminant of a quadratic is the expression inside the radical of the quadratic formula.

$b^{2} - 4 (a c)$

Step 4

Substitute in the values of $a$ , $b$ , and $c$ .

$10^{2} - 4 (1 \cdot - 2)$

Step 5

Evaluate the result to find the discriminant.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $10$ to the power of $2$ .

$100 - 4 (1 \cdot - 2)$

Step 5.1.2

Multiply $- 4 (1 \cdot - 2)$ .

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Step 5.1.2.1

Multiply $- 2$ by $1$ .

$100 - 4 \cdot - 2$

Step 5.1.2.2

Multiply $- 4$ by $- 2$ .

$100 + 8$

Step 5.2

Add $100$ and $8$ .

$108$