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:
+ 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
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.
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.
May 21, 18 09:24 AM
Explore exponential and logarithmic functions with these easy to follow math lessons
New math lessons
Your email is safe with us. We will only use it to inform you about new math lessons.