quadratic formula

Example 02 - Quadratic equation problem

Problem
Determine the equation whose roots are the reciprocals of the roots of the equation 3x2 - 13x - 10 = 0.
 

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
 

Sum and Product of Roots

The quadratic formula
 

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

 

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that
 

$x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a}$   and   $x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a}$

 

where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. The sum of roots x1 + x2 and the product of roots x1·x2 are common to problems involving quadratic equation.
 

Derivation of Quadratic Formula

The roots of a quadratic equation ax2 + bx + c = 0 is given by the quadratic formula
 

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

 

The derivation of this formula can be outlined as follows:

  1.   Divide both sides of the equation ax2 + bx + c = 0 by a.
  2.   Transpose the quantity c/a to the right side of the equation.
  3.   Complete the square by adding b2 / 4a2 to both sides of the equation.
  4.   Factor the left side and combine the right side.
  5.   Extract the square-root of both sides of the equation.
  6.   Solve for x by transporting the quantity b / 2a to the right side of the equation.
  7.   Combine the right side of the equation to get the quadratic formula.

See the derivation below.
 

 
 
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