Quadratic Equations in One Variable

Quadratic Equation
Quadratic equation is in the form
 

$ax^2 + bx + c = 0$

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

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, 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
 

$(x - x_1)(x - x_2) = 0$

Where x1 and x2 are the roots of ax2 + bx + c = 0.
 

Quadratic Formula
For the quadratic equation ax2 + bx + c = 0,
 

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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

$x_1 + x_2 = -\dfrac{b}{a}$

 

Product of roots

$x_1 x_2 = \dfrac{c}{a}$

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

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