### Quadratic equation

Quadratic equation is given in the form:

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

where ‘a’, ‘b’ and ‘c’ are constants and a ≠ 0.

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

Here b^{2}
– 4ac is called as discriminant.

If b^{2} –
4ac > 0, then the equation has two solutions.

If b^{2} –
4ac = 0, then the equation has one solution.

If b^{2} –
4ac < 0, then the equation has no real solutions.

Using quadratic equation,
it is possible to find the value of ‘x’, if ‘a’, ‘b’ and ‘c’ are known. For
example, consider a 2^{nd} order polynomial equation:

y = ax^{2}
+ bx + c.

By using quadratic formula, it is possible to determine the value of ‘x’ data when, y = 0.

If y = 0, then 2^{nd}
order polynomial equation becomes a quadratic equation as,

0 = ax^{2}
+ bx + c when a ≠ 0.

__Example #1__

In an experimental
analysis, for a set of ‘x’ and ‘y’ data, the derived 2^{nd} order
polynomial curve fit equation is:

y = 2x^{2}
– 9x + 10.

Determine the value of ‘x’ when y = 0.

__Solution__

In this a = 2; b =
–9 and c = 10. From this it is possible to determine ‘x’ when y = 0, and the
equation becomes quadratic form: 0 = 2x^{2} – 9x + 10.

Hence ‘x’ can have
two values such as:

So the equation, y = 2x^{2} – 9x + 10 = 0;

at x = 2.0 or x = 2.5.

y = (2 × 2.5^{2})
– (9 × 2.5) + 10 = 0

y = (2 × 2.0^{
2}) – (9 × 2.0) + 10 = 0

__Example #2__

An experimental
result obeys the following 2^{nd} order equation, y = 2x^{2} –
8x + 8. Determine the value of ‘x’ at y = 0.

__Solution__

This equation becomes quadratic equation when y = 0.

2x^{2} –
8x + 8 = 0

In this a = 2, b = –8 and c = 8. The value of ‘x’ can be determined from the quadratic formula as:

So, the value of ‘x’ = 2 when y = 0.

y = 2x^{2}
– 8x + 8 = 0 at x = 2

y = (2 × 2^{2})
– (8 × 2) + 8 = 0

It should be
emphasized that, in the equation: 2x^{2} – 9x + 10 = 0, as given in the
Example #1, the value of (b^{2}
– 4ac) > 0 and hence it has two solutions or two ‘x’ values.

But in Example #2, the quadratic equation, 2x^{2} – 8x + 8 = 0 has only one
solution, since the value of (b^{2} – 4ac) = 0.