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

### Problem #1

*5x ^{2}+8x+3 = 0*

**solution:**

*a* = 5, *b* = 8, *c* = 3

*x _{1,2}* = (-8 ± √(8

^{2}– 4×5×3)) / (2×5) = (-8 ± √(4)) / 10 = (-8 ± 2) / 10

*x*

_{1 }= (-8 + 2) / 10 = -6/10 = -0.6

*x*

_{2 }= (-8 – 2) / 10 = -1

### Problem #2

*4x ^{2 }– 7x + 1 = 0*

**solution:**

*a* = 4, *b* = -7, *c* = 1

*x _{1,2}* = (7 ± √((-7)

^{2}– 4×4×1)) / (2×4) = (7 ± √(49 – 16)) / 8 = (7 ± 5,74) / 8

*x*;

_{1}≈ 1,59*x*

_{2}≈ 0,157### Problem #3

*x ^{2} + 2x + 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 + 2i*

*x*

_{2 }= -1 – 2i### 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).