Explanation:
Quadratic Equation
Quadratic equation is in the form
ax2+bx+c=0
Where
a, b, & c = real-number constants
a & b = numerical coefficient or simply coefficients
a = coefficient of x2
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. 3x2+2x−8=0
2. x2−9=0
3. 2x2+5x=0
4. sin2θ−2sinθ−1=0
5. x−5x−−√+6=0
6. 10x1/3+x1/6−2=0
7. 2lnx−−−√−5lnx−−−√4−7=0
For us to see that the above examples can be treated as quadratic equation, we take example no. 6 above, 10x1/3 + x1/6 - 2 = 0. Let x1/6 = z, thus, x1/3 = z2. The equation can now be written in the form 10z2 + z - 2 = 0, which shows clearly to be quadratic equation.
Roots of a Quadratic Equation
The equation ax2 + bx + c = 0 can be factored into the form
(x−x1)(x−x2)=0
Where x1 and x2 are the roots of ax2 + bx + c = 0.
Quadratic Formula
For the quadratic equation ax2 + bx + c = 0,
x=−b±b2−4ac−−−−−−−√2a
See the derivation of quadratic formula here.
The quantity b2 - 4ac inside the radical is called discriminat.
• If b2 - 4ac = 0, the roots are real and equal.
• If b2 - 4ac > 0, the roots are real and unequal.
• If b2 - 4ac < 0, the roots are imaginary.
Sum and Product of Roots
If the roots of the quadratic equation ax2 + bx + c
= 0 are x1 and x2, then
Sum of roots
x1+x2=−ba
Product of roots
x1x2=ca
You may see the derivation of formulas for sum and product of roots here.