Quadratic equation

Quadratic equation is a second order polynomial with 3 coefficients - a, b, c.

The quadratic equation is given by:

ax2 + bx + c = 0

The solution to the quadratic equation is given by 2 numbers x1 and x2.

We can change the quadratic equation to the form of:

(x - x1)(x - x2) = 0

Quadratic Formula

The solution to the quadratic equation is given by the quadratic formula:

The expression inside the square root is called discriminant and is denoted by Δ:

Δ = b2 - 4ac

The quadratic formula with discriminant notation:

This expression is important because it can tell us about the solution:

  • When Δ>0, there are 2 real roots x1=(-b+√Δ)/(2a) and x2=(-b-√Δ)/(2a).
  • When Δ=0, there is one root x1=x2=-b/(2a).
  • When Δ<0, there are no real roots, there are 2 complex roots:
    x1=(-b+i√)/(2a) and x2=(-b-i√)/(2a).

Problem #1

3x2+5x+2 = 0

solution:

a = 3, b = 5, c = 2

x1,2 = (-5 ± √(52 - 4×3×2)) / (2×3) = (-5 ± √(25-24)) / 6 = (-5 ± 1) / 6

x1 = (-5 + 1)/6 = -4/6 = -2/3

x2 = (-5 - 1)/6  = -6/6 = -1

Problem #2

3x2-6x+3 = 0

solution:

a = 3, b = -6, c = 3

x1,2 = (6 ± √( (-6)2 - 4×3×3)) / (2×3) = (6 ± √(36-36)) / 6 = (6 ± 0) / 6

x1 = x2 = 1

Problem #3

x2+2x+5 = 0

solution:

a = 1, b = 2, c = 5

x1,2 = (-2 ± √(22 - 4×1×5)) / (2×1) = (-2 ± √(4-20)) / 2 = (-2 ± √(-16)) / 2

There are no real solutions. The values are complex numbers:

x1 = -1 + 2i

x2 = -1 - 2i

Quadratic Function Graph

The quadratic function is a second order polynomial function:

f(x) = ax2 + bx + c

 

The solutions to the quadratic equation are the roots of the quadratic function, that are the intersection points of the quadratic function graph with the x-axis, when

f(x) = 0

 

When there are 2 intersection points of the graph with the x-axis, there are 2 solutions to the quadratic equation.

When there is 1 intersection point of the graph with the x-axis, there is 1 solution to the quadratic equation.

When there are no intersection points of the graph with the x-axis, we get not real solutions (or 2 complex solutions).


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