Problem 159 - Algebra and Trigonometry

Question

How is the quadratic formula derived?

Step-by-Step Solution

Answer

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), is derived starting from the standard form of a quadratic equation, \(ax^2 + bx + c = 0\), through a series of algebraic manipulations which largely involved completing the square.

Step 1: Start with the generic quadratic equation

A quadratic equation is given by \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\).

Step 2: Normalize the equation

Divide every term by \(a\) to get the equation in the form \(x^2+\frac{b}{a}x+\frac{c}{a}=0\). This simplifies further to \(x^2+\frac{b}{a}x=-\frac{c}{a}\). The purpose of achieving this format is to simplify the equation for the next step.

Step 3: Complete the square

To complete the square on the term \(\frac{b}{a}x\), take half the coefficient of \(x\), square it and add it to both sides. From this \(\frac{b}{2a}^2\) is derived. The equation turns into \(x^2+\frac{b}{a}x+(\frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a}\). This can be rewritten as \((x + \frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\).

Step 4: Solve for x

Take the square root of both sides to get \(x + \frac{b}{2a}= \pm\frac{\sqrt{b^2-4ac}}{2a}\). Subtract \(\frac{b}{2a}\) from both sides to isolate \(x\). This gives the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).