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.

2020-11-29