**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, 10x^{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 10z^{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} - 4ac inside the radical is called discriminat.

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

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

• If b^{2} - 4ac < 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.