Algebra Examples

$x^{2} + 4 x + 1 = 0$

Step 1

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

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

Step 2

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

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

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 $4$ to the power of $2$ .

$16 - 4 (1 \cdot 1)$

Step 3.1.2

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

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Step 3.1.2.1

Multiply $1$ by $1$ .

$16 - 4 \cdot 1$

Step 3.1.2.2

Multiply $- 4$ by $1$ .

$16 - 4$

Step 3.2

Subtract $4$ from $16$ .

$12$

Step 4

The nature of the roots of the quadratic can fall into one of three categories depending on the value of the discriminant $(Δ)$ :

$Δ > 0$ means there are $2$ distinct real roots.

$Δ = 0$ means there are $2$ equal real roots, or $1$ distinct real root.

$Δ < 0$ means there are no real roots, but $2$ complex roots.

Since the discriminant is greater than $0$ , there are two real roots.

Two Real Roots