Question
Judea solve the quadratic equation x - 96 = 4X by factoring her work is shown below at which step did jodida first make a mistake if at all x equals -12 or x equals 8
Answers
GPT-4o mini
Let's start by solving the equation \( x - 96 = 4x \).
1. **Rearranging the equation**: We can move all terms to one side to set the equation to zero:
\[
x - 96 - 4x = 0
\]
2. **Combining like terms**:
\[
-3x - 96 = 0
\]
3. **Isolating x**:
\[
-3x = 96
\]
\[
x = -32
\]
This leads us to find \( x = -32 \).
Now, let's analyze the steps involved in factoring:
If Judea was trying to factor the equation, she would need to express the quadratic in standard form \( ax^2 + bx + c = 0 \). From the equations above, after moving all terms to one side, we would get:
\[
-3x - 96 = 0 \Rightarrow 3x + 96 = 0 \implies 3x + 96 = 0
\]
If she factored incorrectly and obtained roots of \( x = -12 \) and \( x = 8 \), the original equation would need to have been:
\[
(x + 12)(x - 8) = 0
\]
To check, this would expand back to:
\[
x^2 + 4x - 96 = 0
\]
However, by comparing original steps, it's evident that instead of trying to find such factors, the equation was directly solvable by rearranging terms, yielding a different set of roots. Therefore, the error occurred when Judea assumed that the factors \( (x + 12)(x - 8) = 0 \) related to the original equation \( x - 96 = 4x \).
In conclusion, Judea's first mistake most likely happened when she incorrectly factored or calculated the roots of the equation after moving terms around. Ultimately, the only correct answer for the original equation is \( x = -32 \), not \( x = -12 \) or \( x = 8 \).
1. **Rearranging the equation**: We can move all terms to one side to set the equation to zero:
\[
x - 96 - 4x = 0
\]
2. **Combining like terms**:
\[
-3x - 96 = 0
\]
3. **Isolating x**:
\[
-3x = 96
\]
\[
x = -32
\]
This leads us to find \( x = -32 \).
Now, let's analyze the steps involved in factoring:
If Judea was trying to factor the equation, she would need to express the quadratic in standard form \( ax^2 + bx + c = 0 \). From the equations above, after moving all terms to one side, we would get:
\[
-3x - 96 = 0 \Rightarrow 3x + 96 = 0 \implies 3x + 96 = 0
\]
If she factored incorrectly and obtained roots of \( x = -12 \) and \( x = 8 \), the original equation would need to have been:
\[
(x + 12)(x - 8) = 0
\]
To check, this would expand back to:
\[
x^2 + 4x - 96 = 0
\]
However, by comparing original steps, it's evident that instead of trying to find such factors, the equation was directly solvable by rearranging terms, yielding a different set of roots. Therefore, the error occurred when Judea assumed that the factors \( (x + 12)(x - 8) = 0 \) related to the original equation \( x - 96 = 4x \).
In conclusion, Judea's first mistake most likely happened when she incorrectly factored or calculated the roots of the equation after moving terms around. Ultimately, the only correct answer for the original equation is \( x = -32 \), not \( x = -12 \) or \( x = 8 \).