Exercises. Quadratic equations.

Exercise. Solve the quadratic equation:

4x² + 3 x - 4 = 0

Quadratic equation is a polynomial equation of the second degree. The general form is

a x² + b x + c = 0,

where a ≠ 0.

Solve quadratic equation means to find all the values of x_i, in which will be the equality

a x_i² + b x_i + c = 0.

Necessary to calculate the discriminant for solving the quadratic equation

∆ = b² - 4 a c.

• If the discriminant is positive (∆ > 0), then there are two distinct roots, both of which are real numbers.
• If the discriminant is zero (∆ = 0), then there is exactly one distinct real root, sometimes called a double root (x₁ = x₂).
• If the discriminant is negative (∆ < 0), then there are no real roots.

The roots are given by the quadratic formula

x1,2 =	-b ± √∆
	2 a

Vieta theorem

For the mentioned quadratic equation

x² + px + q = 0

root sum is equal to coefficient p which is drawn with the opposite sign and root’s product is equal to free term q:

x₁ + x₂ = -p;
x₁x₂ = q.