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Chapter 1: Problem 159

How is the quadratic formula derived?

Short Answer

Expert verified
The quadratic formula \(x = [-b \pm \sqrt{(b^2 - 4ac)}]/(2a)\) is derived from the general form of a quadratic equation by dividing the equation by 'a', following up with the process of completing the square, and eventually, rearranging the equation to isolate x.

Step by step solution

01

Write the General Form of Quadratic Equation

A quadratic equation can be written in the form \(ax^2 + bx + c = 0\). This is the standard format, where a, b, and c are constants, and x is the variable we intend to solve for.
02

Divide Equation by 'a'

To simplify the equation, divide every term by the coefficient of \(x^2\), which is 'a'. This results in \(x^2 + (b/a)x + c/a = 0\)
03

Rearrange the Equation

Transfer the constant term to the other side of the equation to facilitate the process of completing the square. This gives us \(x^2 + (b/a)x = -c/a\).
04

Complete the Square

To complete the square on the left side of the equation, add the square of half of the coefficient of x, which is \(((b/2a)^2)\), to both sides. The equation becomes \((x + b/2a)^2 = (b^2/4a^2) - c/a\).
05

Simplify Further

Simplify the right side of the equation to have a common denominator, giving us \((x + b/2a)^2 = (b^2 - 4ac)/4a^2\). Taking square root on both sides we have \(x + b/2a = \pm \sqrt{(b^2 - 4ac)}/(2a)\), which simplifies to \(x = [-b \pm \sqrt{(b^2 - 4ac)}]/(2a)\)

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