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Chapter 1: Problem 161

If you are given a quadratic equation, how do you determine which method to use to solve it?

Short Answer

Expert verified
To decide which method to use for solving a quadratic equation, look at the coefficients. If the equation is easily factorable, use factorization. For equations where the coefficient of \( x^2 \) is 1 and the coefficient of \( x \) is even, use completing the square. If the equation is more complicated, or if a quick solution is wanted regardless of simplicity, use the quadratic formula.

Step by step solution

01

Understanding the Features of the Quadratic Equation

Firstly, understand the features of the quadratic equation. A standard quadratic equation is of the form \( ax^2 + bx + c = 0 \). The coefficients \( a \), \( b \), and \( c \) will help to determine the best method for solving the equation.
02

Choosing the Factorization Method

If the equation can easily be factored, this would be the best method to use. For instance, this is typically the case when \( a = 1 \), and \( b \) and \( c \) are small integers.
03

Choosing the Completing the Square Method

Completing the square is a method commonly used when the coefficient \( a = 1 \) and the coefficient \( b \) is an even number. This method transforms the equation into a form that allows for straightforward solution by taking square roots.
04

Choosing the Quadratic Formula Method

When the equation is in a more complicated form, it is typically fastest to use the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). This formula derives from the completing the square method and provides a solution for any equation of the form \( ax^2 + bx + c = 0 \), regardless of the coefficients.

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