My Math Section
The following symbols are used in articles of the math section. Just make sure you understand what they mean.
"^" means "the power of", 2^2 means 2 to the second power which equals 4.
"Sqrt" means "the square root of", Sqrt(9) means the square root of 9 which is 3
"(+,-)" means "plus or mines",4 (+,-) 6 means 4 plus or mines 6
"/" means divide
"*" means multiply
"=)" a sign I invented, it means "next'
How does the Quadratic formula works
Well, I know that the formula of the equation: "ax^2 + bx + c = 0" is "[-b (+,-) Sqrt(b^2 - 4ac)] / 2a"
But most people ask "How does it work?", I am writing it down here so you can show off in front of other people that doesn't know anything about it. ^_^
Formula to Equation:
[-b (+,-) Sqrt(b^2 - 4ac)] / 2a = x
=) 2ax+b = (+,-) Sqrt(b^2 - 4ac) move the 2a and the -b to the other side
=) [2ax + b]^2 = [ (+,-) Sqrt(b^2 - 4ac)]^2 raise both sides to the second power so the Sqrt sign and the plus or mines sign canceled
=) [2ax + b] * [2ax + b] = b^2 - 4ac simplify both side
=) [2ax + b] * [2ax + b] - (b^2 - 4ac) = 0 move the right side to the left
=) [4 * a^2 * x^2 + 4axb + b^2] - b^2 + 4ac = 0 using the expanding method and the distributive property
=) 4 * a^2 * x^2 + 4axb + 4ac = 0 cancel the positive b^2 and the negative b^2
=) 4 (a^2*x^2 + axb + ac) = 0 pull out the common factor 4
=) a^2*x^2 + axb + ac = 0 divide both side by 4
=) a (a*x^2 + xb + c) = 0 pull out the common factor a
=) ax^2 + xb + c = 0 divide each side by a
=) ax^2 + bx +c = 0 If that wasn't obvius enough, I put the b in front of the x
How does the Quadratic Formula works (second part)
Now, equation to formula is a bit more complicated, you just need to understand that our main goal is to actually FACTOR the equation using the function : "a^2 + 2ab + b^2 = (a+b)^2".
ax^2 + bx +c = 0
=) a(ax^2 + bx +c) = 0 Multiply both sides by an "a"
=) (ax)^2 + abx + ac = 0 Simplify
=) 2 [ (ax)^2 + abx + ac ] = 0 Multiply both sides by 2
=) 2*(ax)^2 + 2axb + 2ac = 0 Simplify
=) 2 [2*(ax)^2 + 2axb + 2ac] = 0 Multiply both sides by 2 again
=) 4*(ax)^2 + 2*(2axb) + 4ac = 0 Using the distributive property
=) (2ax)^2 + 2(2ax)b + 4ac = 0 Simplify
Now, think that "2ax" is "a", "b" is "b", we need another "b^2" in order to have the equation "a^2 + 2ab + b^2"
=) (2ax)^2 + 2(2ax)b + b^2 - b^2 + 4ac = 0 I added the b^2 and mines it, so the equation won't change
=) (2ax + b)^2 -b^2 + 4ac = 0 e have a completed model of factoring, so factor it!
=) (2a x+ b)^2 = b^2 - 4ac move the -b^2 + 4ac to the other side, our goal now is to find x
=) (+,-)(2ax + b) = Sqrt(b^2 - 4ac) add the Square root sign on both sides
=) 2ax + b = (+,-) Sqrt(b^2 - 4ac) move the plus or mines sign to the pther side
=) 2ax = -b (+,-) Sqrt(b^2 - 4ac) move the b to the other side
=) x = [-b (+,-) Sqrt(b^2 - 4ac)] / 2a divide both sides by 2a and now we have completed this whole process, and that's how the quadratic formula formed and worked.