Proof of the quadratic formula
The following is a proof of the quadratic formula. It will show you how the quadratic formula that is widely used was developed.
The proof is done using the standard form of a quadratic equation and solving the standard form by completing the square
Start with the the standard form of a quadratic equation:
ax
^{2} + bx + c = 0
Divide both sides of the equation by a so you can complete the square
Subtract c/a from both sides
Complete the square:
The coefficient of the second term is b/a
Divide this coefficient by 2 and square the result to get (b/2a)
^{2}
Add (b/2a)
^{2} to both sides:
Since the left side of the equation right above is a perfect square, you can factor the left side by using the coefficient of the first term (x) and the base of the last term(b/2a)
Add these two and raise everything to the second.
Then, square the right side to get (b
^{2})/(4a
^{2})
Get the same denominator on the right side:
Now, take the square root of each side:
Simplify the left side:
Rewrite the right side:
Subtract b/2a from both sides:
Adding the numerator and keeping the same denominator, we get the quadratic formula:
The + - between the b and the square root sign means plus or negative. In other words, most of the time, you will get two answers when using the quadratic formula.