# Quadratic equations. The discriminant

If we have quadratic equation $ax^2+bx+c=0$, then the discriminant of the quadratic equation is the number D and it is found by formula.

$$D=b^2-4ac$$Then the solutions $(x_1 \ \text{and} \ x_2)$ of the quadratic equation we will find with this formula:

$$\displaystyle x_{1,2}=\frac{-b \pm \sqrt{D}}{2a}$$We have quadratic equation like this: $2x^2-3x-2=0$. Solve it with discriminant formula.

$$x_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{3\pm \sqrt{(-3)^2-4\cdot 2 \cdot (-2)}}{2 \cdot 2}$$ $$x_1=2 \ \text{and} \ x_2=-\frac{1}{2}$$## Number of solutions

If we solve quadratic equation with discriminant and we get $D > 0$, then the quadratic equation has two distinct real solutions. If $D = 0$, then the quadratic equation has one real solution. If $D < 0$, then the quadratic equation has no real solutions.

How many real solutions does equation $2x^2+6x+5=0$ have?

$$D=b^2-4ac=6^2-4\cdot 2 \cdot 5=-4<0$$Answer: this quadratic equation has no real solutions.

## Wanna check your skills?

A quiz is based on this article and has 5 questions.

### Quiz

2020-11-29

## All categories

## New content

- Circle: arc, sector and segment (Geometry)
- Circle: radius, diameter, circumference and area (Geometry)
- The law of cosines and examples (Geometry)
- The law of sines and examples (Geometry)
- Rewriting radicals into exponential form (Algebra)
- Infinite geometric sequence sum formula and examples (Algebra)
- Geometric sequence formulas and examples (Algebra)
- Arithmetic sequence formulas and examples (Algebra)