Real numbers: a,b,c,d,p,q,r,s

Solutions: $ x_{1},x_{2},y_{1},y_{2},y_{3}$

Linear Equation in One Variable

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Quadratic Equation

In Qudratic formula $(b^2- 4ac)$ is called the Discriminant, where a,b and c are real numbers, a $\neq$ 0.It is represented by D.

• When D is positive, then the roots are real and unequal.
• When D is zero, then roots are real and equal.
• When D is negative, then roots are unequal and imaginary.
• When D is positive and perfect square, then the roots are real,rational and unequal.
• When D is positive but not perfect square, then roots are real, irrational and unequal.
• When D is perfect square and a or b is irrational, then roots are irrational.

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Viete's Formulas

$1.~ a^2+bx=0 \\ \text{Then, } x_{1}=0,x_{2}=-\dfrac{b}{a} \\$ $2.~ a^2+c=0 \\ \text{Then, } x_{1,2}=\pm \sqrt{\dfrac{-c}{a}} \\$

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Cubic Equation

$ where, \ r = \sqrt[\Large3]{-\dfrac{q}{2} + \sqrt{\left(\dfrac{q}{2}\right) ^2 + \left(\dfrac{p}{2}\right) ^3}} \\ s = \sqrt[\Large3]{-\dfrac{q}{2} + \sqrt{\left(\dfrac{q}{2}\right) ^2 - \left(\dfrac{p}{2}\right) ^3}} $