Finite Math Examples

- 5 x^{2} + 10 x - 16

$- 5 x^{2} + 10 x - 16$

Step 1

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

b^{2} - 4 (a c)

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

Step 2

Substitute in the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

10^{2} - 4 (- 5 \cdot - 16)

$10^{2} - 4 (- 5 \cdot - 16)$

Step 3

Evaluate the result to find the discriminant.

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Step 3.1

Simplify each term.

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Step 3.1.1

Raise

10

$10$ to the power of

2

$2$ .

100 - 4 (- 5 \cdot - 16)

$100 - 4 (- 5 \cdot - 16)$

Step 3.1.2

Multiply

- 4 (- 5 \cdot - 16)

$- 4 (- 5 \cdot - 16)$ .

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Step 3.1.2.1

Multiply

- 5

$- 5$ by

- 16

$- 16$ .

100 - 4 \cdot 80

$100 - 4 \cdot 80$

Step 3.1.2.2

Multiply

- 4

$- 4$ by

80

$80$ .

100 - 320

$100 - 320$

100 - 320

$100 - 320$

100 - 320

$100 - 320$

Step 3.2

Subtract

320

$320$ from

100

$100$ .

- 220

$- 220$

- 220

$- 220$

Step 4

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant

(Δ)

$(Δ)$ :

Δ > 0

$Δ > 0$ means there are

2

$2$ distinct real roots.

Δ = 0

$Δ = 0$ means there are

2

$2$ equal real roots, or

1

$1$ distinct real root.

Δ < 0

$Δ < 0$ means there are no real roots, but

2

$2$ complex roots.

Since the discriminant is less than

0

$0$ there are no real roots. Instead, there are two complex roots.

Two Complex Roots