Question

When trying to sell the quite a lot of equation x + 4x - 5 = 7 Jamal and George each provided to work in the tables each said they use the zero product property to solve the equation after step 2 explain how you know the know the correct student solution is accurate and how you know the incorrect student solution is inaccurate step two right quarterback of quadratic equation as a product of two factors x^2 + 4x - 5 = 7

Answer 1

Let's first clarify and solve the given equation step-by-step, and then analyze the students' approaches based on the zero product property.

Given Equation: \[ x + 4x - 5 = 7 \]

Step 1: Simplify the equation Combine like terms: \[ 5x - 5 = 7 \]

Step 2: Move all terms to one side Add 5 to both sides: \[ 5x = 12 \]

Now, divide by 5: \[ x = \frac{12}{5} \]

Since both Jamal and George mentioned they used the zero product property, it suggests they may have altered the original equation despite the fact that it does not explicitly present as a product of factors. Let's transform the equation into a standard quadratic form.

Transforming to Quadratic Form: To create a quadratic equation from the original linear equation, we can rewrite the equation as follows:

\[ x + 4x - 5 - 7 = 0 \]

Combine like terms: \[ 5x - 12 = 0\]

This is actually a linear equation, but to use the zero product property, we can create a quadratic equation like this (although it may not apply directly in this context): \[ x^2 + 4x - 5 = 0 \]

Step 3: Factor the Quadratic To factor \( x^2 + 4x - 5 = 0 \), we find two numbers that multiply to -5 (the constant) and add to 4 (the coefficient of x). These numbers are 5 and -1.

Thus, we can factor it as: \[ (x + 5)(x - 1) = 0 \]

Step 4: Apply the Zero Product Property By the zero product property: Either \( x + 5 = 0 \) or \( x - 1 = 0 \).

Solving these gives us:

1. \( x = -5 \)
2. \( x = 1 \)

Evaluating the solutions:

1. The correct solution from the original equation was \( x = \frac{12}{5} \).
2. The solutions \( x = -5 \) and \( x = 1 \) generated from the quadratic equation do not satisfy the original equation as they were created from a different setup.

Conclusion: The correct solution is accurate as \( \frac{12}{5} \) is derived directly from manipulating the linear equation \( 5x - 12 = 0\). The incorrect results from manipulating the quadratic equation \( x^2 + 4x - 5 = 0 \) are inaccurate because they do not apply to the original context of the problem in which there was no zero product situation applicable without appropriate manipulation of the original terms.

Thus, while Jamal and George may have validated their individual interpretations using the zero product property, their conclusions were inaccurate based on the initial linear equation and context.