What is quadratic formula? Prove and drive
Answers
Answer:
QUADRATIC FORMULA: -b+-√D/2a...
i hope it help u
Step-by-step explanation:
Derivation of Quadratic Formula
A Quadratic Equation looks like this:
Quadratic Equation: ax^2 + bx + c = 0
And it can be solved using the Quadratic Formula:
Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a
That formula looks like magic, but you can follow the steps to see how it comes about.
1. Complete the Square
ax2 + bx + c has "x" in it twice, which is hard to solve.
But there is a way to rearrange it so that "x" only appears once. It is called Completing the Square (please read that first!).
Our aim is to get something like x2 + 2dx + d2, which can then be simplified to (x+d)2
So, let's go:
Start with ax^2 + bx + c=0
Divide the equation by a x^2 + bx/a + c/a = 0
Put c/a on other side x^2 + bx/a = -c/a
Add (b/2a)2 to both sides x^2 + bx/a + (b/2a)^2 = -c/a + (b/2a)^2
The left hand side is now in the x2 + 2dx + d2 format, where "d" is "b/2a"
So we can re-write it this way:
"Complete the Square" (x+b/2a)^2 = -c/a + (b/2a)^2
Now x only appears once and we are making progress.
2. Now Solve For "x"
Now we just need to rearrange the equation to leave "x" on the left
Start with (x+b/2a)^2 = -c/a + (b/2a)^2
Square root (x+b/2a) = (+-) sqrt(-c/a+(b/2a)^2)
Move b/2a to right x = -b/2a (+-) sqrt(-c/a+(b/2a)^2)
That is actually solved! But let's simplify it a bit:
Multiply right by 2a/2a x = [ -b (+-) sqrt(-(2a)^2 c/a + (2a)^2(b/2a)^2) ] / 2a
Simplify: x = [ -b (+-) sqrt(-4ac + b^2) ] / 2a
Which is the Quadratic formula we all know and love:
Quadratic Formula: x = [ -b (+-) sqrt(b^2 - 4ac) ] / 2a