# Quadratic Equations in One Variable

**Quadratic Equation**

Quadratic equation is in the form

Where

*a*, *b*, & *c* = real-number constants

*a* & *b* = numerical coefficient or simply coefficients

*a* = coefficient of *x*^{2}

*b* = coefficient of *x*

*c* = constant term or simply constant

*a* cannot be equal to zero while either *b* or *c* can be zero

**Examples of Quadratic Equation**

Some quadratic equation may not look like the one above. The general appearance of quadratic equation is a second degree curve so that the degree power of one variable is twice of another variable. Below are examples of equations that can be considered as quadratic.

1. $3x^2 + 2x - 8 = 0$

2. $x^2 - 9 = 0$

3. $2x^2 + 5x = 0$

4. $\sin^2 \theta - 2\sin \theta - 1 = 0$

5. $x - 5\sqrt{x} + 6 = 0$

6. $10x^{1/3} + x^{1/6} - 2 = 0$

7. $2\sqrt{\ln x} - 5\sqrt[4]{\ln x} - 7 = 0$

For us to see that the above examples can be treated as quadratic equation, we take example no. 6 above, 10*x*^{1/3} + *x*^{1/6} - 2 = 0. Let *x*^{1/6} = *z*, thus, *x*^{1/3} = *z*^{2}. The equation can now be written in the form 10*z*^{2} + *z* - 2 = 0, which shows clearly to be quadratic equation.

**Roots of a Quadratic Equation**

The equation *ax*^{2} + *bx* + *c* = 0 can be factored into the form

Where *x*_{1} and *x*_{2} are the roots of *ax*^{2} + *bx* + *c* = 0.

**Quadratic Formula**

For the quadratic equation *ax*^{2} + *bx* + *c* = 0,

See the derivation of quadratic formula here.

The quantity *b*^{2} - 4*ac* inside the radical is called discriminat.

• If *b*^{2} - 4*ac* = 0, the roots are real and equal.

• If *b*^{2} - 4*ac* > 0, the roots are real and unequal.

• If *b*^{2} - 4*ac* < 0, the roots are imaginary.

**Sum and Product of Roots**

If the roots of the quadratic equation *ax*^{2} + *bx* + *c* = 0 are *x*_{1} and *x*_{2}, then

Sum of roots

Product of roots

You may see the derivation of formulas for sum and product of roots here.