Lesson PROOF of quadratic formula by completing the square

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PROOF of quadratic formula by completing the square


This lesson will prove that quadratic equations can be solved by "completing the square", and I will show you how it is done!

Consider the quadratic equation

a%2Ax%5E2%2Bb%2Ax%2Bc = 0                 (1)

assuming coefficients  a,  b  and  c  are real numbers.

Use  "completing the square".  Add and subtract  b%5E2%2F%284%2Aa%29  in the left side.  Your equation takes the form
a%2Ax%5E2%2Bb%2Ax%2Bc + %28b%5E2%2F%284%2Aa%29%29 - %28b%5E2%2F%284%2Aa%29%29 = 0.         (2)

Regroup the terms and factor for a:

a%2A%28x%5E2%2Bb%2Fa%2Ax%2Bb%5E2%2F%284%2Aa%5E2%29%29 - b%5E2%2F%284%2Aa%29 + c = 0.         (3)
The expression in parentheses is the complete square   %28x%5E2%2Bb%2Fa%2Ax%2Bb%5E2%2F%284%2Aa%5E2%29%29 = %28x%2Bb%2F%282%2Aa%29%29%5E2.
Therefore,  the equation  (3)  becomes

a%2A%28x%2Bb%2F%282%2Aa%29%29%5E2 - b%5E2%2F%284%2Aa%29 + c = 0,     or

a%2A%28x%2Bb%2F%282%2Aa%29%29%5E2 = b%5E2%2F%284%2Aa%29 - c,             or

a%2A%28x%2Bb%2F%282%2Aa%29%29%5E2 = %28b%5E2-4%2Aa%2Ac%29%2F%284%2Aa%29.

Multiply both sides by  1%2Fa.  You will get

%28x%2B%28b%2F%282%2Aa%29%29%29%5E2 = %28b%5E2-4%2Aa%2Ac%29%2F%284%2Aa%5E2%29.         (4)

An expression   d+=+b%5E2-4%2Aa%2Ac   is called  the discriminant  of the quadratic equation  (1).

Assuming the discriminant is positive or zero (no negative), take the square root for both sides.  Note we are still in the area of real numbers.

x%2Bb%2F%282%2Aa%29=sqrt%28%28b%5E2-4%2Aa%2Ac%29%2F%284%2Aa%5E2%29%29   or   x%2Bb%2F%282%2Aa%29=-sqrt%28%28b%5E2-4%2Aa%2Ac%29%2F%284%2Aa%5E2%29%29

x = %28-b%29%2F%282%2Aa%29%2Bsqrt%28b%5E2-4%2Aa%2Ac%29%2F%282%2Aa%29     or   x = %28-b%29%2F%282%2Aa%29-sqrt%28b%5E2-4%2Aa%2Ac%29%2F%282%2Aa%29

x = %28-b%2B-sqrt%28b%5E2-4%2Aa%2Ac%29%29%2F%282%2Aa%29                   (5)

Conclusion:

1)  If the discriminant of the quadratic equation  (1)  with real coefficients  a,  b  and  c  is positive,
      then the equation has two different real roots given by formula  (5).

2)  If the discriminant of the quadratic equation  (1)  is equal to zero,
      then the equation has one real root given by formula  (5).

3)  If the discriminant of the quadratic equation  (1)  is negative,
      then the equation has no real roots.
      It follows from formula  (4)  because the left side as a square of the real number can not be negative.

Note for the area of complex numbers.

Let us consider the quadratic equation  (1)  in the area of complex numbers.
This means that coefficients  a,  b  and  c  are complex numbers and we search a solution among complex numbers.
Then the equation always has two roots in the area of complex numbers.
These roots are given by formula  (4).
If the discriminant is zero,  then the two roots are actually equal each to other.


On solving quadratic equations see also other the lessons in this site
    - Introduction into Quadratic Equations

    - HOW TO complete the square - Learning by examples
    - HOW TO solve quadratic equation by completing the square - Learning by examples
    - Solving quadratic equations without quadratic formula
    - Who is who in quadratic equations
    - Using Vieta's theorem to solve quadratic equations and related problems

    - Find a number using quadratic equations
    - Find an equation of the parabola passing through given points
    - Problems on quadratic equations to solve them using discriminant
    - Relative position of a straight line and a parabola on a coordinate plane
    - Advanced minimax problems to solve them using the discriminant principle

    - Using quadratic equations to solve word problems
    - Word problems on engineering constructions of parabolic shapes
    - Challenging word problems solved using quadratic equations
    - Business-related problems to solve them using quadratic equations
    - Rare beauty investment problem to solve using quadratic equation
    - HOW TO solve the problem on quadratic equation mentally and avoid boring calculations
    - Entertainment problems on quadratic equations
    - Prime quadratic polynomials with real coefficients

    - Problem on a projectile moving vertically up and down
    - Problem on an arrow shot vertically upward
    - Problem on a ball thrown vertically up from the top of a tower
    - Problem on a toy rocket launched vertically up from a tall platform

    - Problems on the area and the dimensions of a rectangle
    - Problems on the area and the dimensions of a rectangle surrounded by a strip
    - Problems on a circular pool and a walkway around it

    - OVERVIEW of lessons on solving quadratic equations

Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.


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