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Chapter 1: Problem 122

How is the quadratic formula derived?

Short Answer

Expert verified
The quadratic formula \( x = [-b 卤 sqrt(b^2 - 4ac)] / 2a \) is derived by completing the square on a general quadratic equation and then applying square root property and simplifying.

Step by step solution

01

Write the Quadratic Equation

Write down the general quadratic equation \( ax^2 + bx + c = 0 \)
02

Make 'a' equal to 1

Divide all terms by 鈥榓鈥� to make coefficient of square term equal to 1. \( x^2 + (b/a)x + c/a = 0 \)
03

Move the Constant on the Other Side

Move the constant \( c/a \) to another side of the equation to make it \( x^2 + (b/a)x = -c/a \)
04

Complete the Square

Complete the square on the left side of the equation and balance it by adding the same value to the right side. \( x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2 \)
05

Application of Square Root Property

Apply square root property to solve for \( x \). \( x = [-b 卤 sqrt{ (b^2 - 4ac) / (4a^2) }] / 2a \)
06

Simplify

Simplify the above expression to obtain the quadratic formula. \( x = [-b 卤 sqrt{b^2 - 4ac} ] / 2a \)

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Most popular questions from this chapter

Solve each rational inequality in Exercises \(29-48,\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{4-2 x}{3 x+4} \leq 0 $$

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Explain how to add complex numbers. Provide an example with your explanation.
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