(a+b)³ + (b+c)³ + (c+a )³ -3 (a+b)(b+c) (c+a)=2(a³+b³+c³) prove it.
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Answers
Answer:
Step-by-step explanation:
a²+b²+2ab=(a+b)²
a²+b²-2ab=(a-b)²
a²-b²=(a+b)(a-b)
a³+b³=(a+b)(a²+b²-ab)
a³-b³=(a-b)(a²+b²+ab)
a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)
a³+b³+3ab(a+b)=(a+b)³
Step-by-step explanation:
Answer:
The Quadratic Equation has many uses in Engineering and Mathematics. It can be used to represent Complex Numbers which are based on the Square Root of -1 : Ö-1 which has no meaning in reality as both 1 x 1 and -1 x -1 yield 1! It can however be used to represent a two dimensional Cartesian number with the real component being the X-axis, and the imaginary component Ö-1 being the Y-Axis. The Square Root of -1 is known a j by Engineers and i by Mathematicians. With j x j being -1 by definition. One use for Complex Numbers is Robotic Servo-motor theory. Cartesian geometry is named after the French Philosopher and Mathematician René Descartes (1596-1650). He also deduced the Philosophical Axiom: 'I think there for I am' (Latin: Cogito ergo sum; French: Je pense, donc je suis).
-b ± Öb² - 4ac
x = 2a Multiply both sides by 2a giving:
2ax = -b ± Öb² - 4ac Add b to both sides giving:
2ax + b = ± Öb² - 4ac Square both sides, remembering (+x)² and (-x)² both
give x², thus giving:
(2ax + b)² = b² - 4ac Multiplying out the Algebraic square gives:
(2ax + b)(2ax + b) = b² - 4ac
4a²x² + 2.2axb + b² = b² - 4ac
Note . is short for Multiply. Next subtract b² from both sides (note
the b² on each side cancels the other out) to give:
4a²x² + 4axb = - 4ac Divide both sides by 4a giving:
ax² + xb = - c Add c to both sides giving:
ax² + bx + c = 0 Which gives the final proof!
Of course the original Quadratic Equation had to be solved in the reverse order to this. So some mathematician had to think to multiply by 4a and add b² to both sides. This isn't as difficult as it might appear as this mathematician would know a Algebraic square would be needed to isolate x, remember 2x2 = 4 and bxb = b². If you ever have to remember the Quadratic Equation for an exam knowing how it is derived will help jog your memory and allow you to check if you remembered it correctly.
The simplest form of Algebra was originally developed by an Arabic scholar in Baghdad in the early 800's. He was called Al-Khowarizmi, the word algorithm is derived from his name. The word algebra is derived from the first words of his most well known book Al Jabr Wa'l Muqabalab. His work used earlier concepts such as Hindu symbols, Mesopotamian mathematics and Euclid's geometry. '0' and the highly important decimal system were developed in India. Try multiplying two Roman Numerals together!