# Quadratic Equation

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

The quadratic equation is given by:

*ax*^{2 }+* bx *+* c* = 0

The solution to the quadratic equation is given by 2 numbers x_{1} and x_{2}.

We can change the quadratic equation to the form of:

(*x *-* x*_{1})(*x *-* x*_{2}) = 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 = *b*^{2} - 4*ac*

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 x
_{1}=(-b+√Δ)/(2a)
and x_{2}=(-b-√Δ)/(2a)_{.}
- When Δ=0, there is one root x
_{1}=x_{2}=-b/(2a)_{.}
- When Δ<0, there are no real roots, there are 2 complex roots:

x_{1}=(-b+i√-Δ)/(2a)
and x_{2}=(-b-i√-Δ)/(2a)_{.}

### Problem #1

3*x*^{2}+5*x*+2 = 0

#### solution:

*a* = 3, *b* = 5, *c* = 2

*x*_{1,2} = (-5 ± √(5^{2} -
4×3×2)) / (2×3) = (-5 ± √(25-24)) / 6 = (-5 ± 1) / 6

*x*_{1 }= (-5 + 1)/6 = -4/6 = -2/3

*x*_{2 }= (-5 - 1)/6 = -6/6 = -1

### Problem #2

3*x*^{2}-6*x*+3 = 0

#### solution:

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

*x*_{1,2} = (6 ± √( (-6)^{2}
- 4×3×3)) / (2×3) = (6 ± √(36-36)) / 6 = (6 ± 0) / 6

*x*_{1 }=* x*_{2 }=
1

### Problem #3

*x*^{2}+2*x*+5 = 0

#### solution:

*a* = 1, *b* = 2, *c* = 5

*x*_{1,2}
= (-2 ± √(2^{2} -
4×1×5)) / (2×1) = (-2 ± √(4-20)) / 2 = (-2 ± √(-16)) / 2

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

*x*_{1 }= -1 + 2*i*

*x*_{2 }= -1 - 2*i*

### Quadratic Function Graph

The quadratic function is a second order polynomial function:

*f*(*x*) = *ax*^{2}* + 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).

## See also