Problem 362 - Elementary Algebra

Question

Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

Step-by-Step Solution

Answer

An example is \[2x^2 + 4x + 2 = 0\], simplified to\[x^2 + 2x + 1 = 0\] with solutions \[x = -1\].

Step 1: Define a general quadratic equation

A quadratic equation has the form \[ax^2 + bx + c = 0\]. To ensure the equation has a Greatest Common Factor (GCF), choose coefficients that share a common divisor greater than 1.

Step 2: Choose coefficients with a GCF

Select coefficients for the quadratic equation such that they share a common factor. For example, \[2x^2 + 4x + 2 = 0\] where the GCF of 2, 4, and 2 is 2.

Step 3: Simplify the quadratic equation

Divide the entire equation by the GCF. For \[2x^2 + 4x + 2 = 0\], divide each term by 2: \[x^2 + 2x + 1 = 0\].

Step 4: Solve the simplified quadratic equation

Solve the equation \[x^2 + 2x + 1 = 0\] by factoring. The factors are \[(x+1)(x+1) = 0\].

Step 5: Find the solutions

Set each factor to zero: \[x+1 = 0\]. Solve for x to find the solutions: \[x = -1\].

Step 6: Verify that none of the solutions are zero

Both solutions to the simplified equation \[x+1 = 0\] are \[x = -1\], which is not zero.

Key Concepts

Greatest Common Factor

When dealing with quadratic equations, understanding the concept of the Greatest Common Factor (GCF) can simplify your computations. The GCF is the largest number that divides two or more numbers without leaving a remainder. For example, consider the numbers 2, 4, and 2. The GCF of these numbers is 2 because it is the largest number that can divide each term without a remainder.

Why is GCF Important in Quadratic Equations?

• Reduces the quadratic equation to a simpler form.
• Makes factoring easier.
• Helps to identify and solve the equation more efficiently.

In quadratic equations, if you choose coefficients that share a GCF, you can divide the entire equation by this GCF. This makes the equation simpler and easier to factor and solve. For instance, in the equation \[2x^2 + 4x + 2 = 0\], the coefficients are 2, 4, and 2. By dividing each term by the GCF of 2, the equation simplifies to \[x^2 + 2x + 1 = 0\].

Factoring

Factoring is a crucial step in solving quadratic equations. It involves breaking down an equation into simpler