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Chapter 9: Problem 65

What is the quadratic formula and why is it useful?

Short Answer

Expert verified
The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\). It is used to find the roots of a quadratic equation. It is particularly useful as it can solve any quadratic equation, regardless of how it may initially appear, including cases where factorization or completing the square might be challenging.

Step by step solution

01

Define the Quadratic Formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\). This formula is used to find the roots \(x\), of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients.
02

Understand the Components of the Formula

The symbols \(a\), \(b\), and \(c\) in the quadratic formula come from the quadratic equation. Specifically, \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant. The symbols \(\pm\) means that there will be two solutions, one using addition and another using subtraction, as defined by the square root term under the radical sign \( \sqrt{{b^2-4ac}} \).
03

Explain the Usage of the Quadratic Formula

The quadratic formula is useful because it allows for solving any quadratic equation, regardless of how it may initially appear. That includes equations where it might not be immediately obvious how to proceed by factorization or completing the square. Additionally, it can be used to determine the type of roots a quadratic equation has, depending on the value of the term under the square root, known as the 'discriminant'. When \(b^2-4ac\) is greater than 0, there are two real roots. When it equals 0, there is a repeated real root. If \(b^2-4ac\) is less than 0, the roots are complex.

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