Determine the Nature of the Roots Using the Discriminant x^2-12x+40=4

$x^{2} - 12 x + 40 = 4$

Step 1

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

Subtract $4$ from both sides of the equation.

$x^{2} - 12 x + 40 - 4 = 0$

Subtract $4$ from $40$ .

$x^{2} - 12 x + 36 = 0$

Step 2

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

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

Step 3

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

${(- 12)}^{2} - 4 (1 \cdot 36)$

Step 4

Evaluate the result to find the discriminant.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$144 - 4 (1 \cdot 36)$

Multiply $- 4 (1 \cdot 36)$ .

Multiply $36$ by $1$ .

$144 - 4 \cdot 36$

Multiply $- 4$ by $36$ .

$144 - 144$

Subtract $144$ from $144$ .

$0$

Step 5

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 equal to $0$ , there are two equal roots, or one distinct real root.

One Real Root