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Chapter 0: Problem 154

How is the quadratic formula derived?

Short Answer

Expert verified
The quadratic formula \(x = [-b \卤sqrt{(b^2 - 4ac)}/(2a)\] is derived by the method of completing the square.

Step by step solution

01

Start with the standard form of the quadratic equation

The standard form of a quadratic equation is \(ax^2 +bx +c = 0\). This will be the starting point of the derivation.
02

Express the equation in terms of 'x'

Divide every term by 'a' to get the equation in the form of \(x^2 + (b/a)x + c/a = 0\). Then, isolate the x-terms and constant term to get \(x^2 + (b/a)x = -c/a\).
03

Complete the square in the equation

To complete the square, add \((b/2a)^2\) to both sides of the equation to get \(x^2 +(b/a)x + (b/2a)^2 = (b/2a)^2 - c/a\). This can be expressed as \((x+b/2a)^2 = (b^2 - 4ac)/4a^2\).
04

Find the square root of both sides

Taking square root on both sides results in \(x + b/2a = \卤sqrt{(b^2 - 4ac)}/2a\).
05

Solve for 'x'

Finally, solve for 'x' to get \(x = [-b \卤sqrt{(b^2 - 4ac)}/(2a)\], which is known as the quadratic formula.

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