Question
Jedida solved the quadratic equation x2−96=4x
by factoring. Her work is shown below. Explain what step she made a mistake and what she did wrong.
(2 points)
by factoring. Her work is shown below. Explain what step she made a mistake and what she did wrong.
(2 points)
Answers
GPT-4o mini
To solve the quadratic equation \( x^2 - 96 = 4x \), we first need to rearrange it into standard form. Moving all terms to one side, we get:
\[
x^2 - 4x - 96 = 0.
\]
Now let’s analyze stating that Jedida solved it by factoring. The equation can be factored into:
\[
(x - a)(x + b) = 0,
\]
where \( a \) and \( b \) are numbers that we need to find.
To factor this equation, we need two numbers that multiply to \(-96\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)).
The pair of numbers that satisfies this condition is \( 8 \) and \(-12\):
\[
x^2 - 4x - 96 = (x - 12)(x + 8) = 0.
\]
When we attempt to solve the equation, we set each factor to zero:
\[
x - 12 = 0 \quad \Rightarrow \quad x = 12,
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8.
\]
Now, if Jedida made a mistake, it could have been in the following areas:
1. **Miscalculating the factors**: If she did not identify the correct pairs of numbers that multiply to \(-96\) and add to \(-4\).
2. **Incorrectly setting the factors to zero**: Even if she factored correctly, if she misapplied the zero-product property or made errors in solving the resulting equations, then she would get the wrong solutions.
In conclusion, to pinpoint Jedida's mistake, we need to look at her specific steps in factoring the equation. If she failed to properly determine the pairs of factors or miscalculated them, that would be her error. Without seeing her exact work, we can only generalize that the mistake lies in either identifying the correct factors or in misapplying algebraic principles while solving for \(x\).
\[
x^2 - 4x - 96 = 0.
\]
Now let’s analyze stating that Jedida solved it by factoring. The equation can be factored into:
\[
(x - a)(x + b) = 0,
\]
where \( a \) and \( b \) are numbers that we need to find.
To factor this equation, we need two numbers that multiply to \(-96\) (the constant term) and add up to \(-4\) (the coefficient of \(x\)).
The pair of numbers that satisfies this condition is \( 8 \) and \(-12\):
\[
x^2 - 4x - 96 = (x - 12)(x + 8) = 0.
\]
When we attempt to solve the equation, we set each factor to zero:
\[
x - 12 = 0 \quad \Rightarrow \quad x = 12,
\]
\[
x + 8 = 0 \quad \Rightarrow \quad x = -8.
\]
Now, if Jedida made a mistake, it could have been in the following areas:
1. **Miscalculating the factors**: If she did not identify the correct pairs of numbers that multiply to \(-96\) and add to \(-4\).
2. **Incorrectly setting the factors to zero**: Even if she factored correctly, if she misapplied the zero-product property or made errors in solving the resulting equations, then she would get the wrong solutions.
In conclusion, to pinpoint Jedida's mistake, we need to look at her specific steps in factoring the equation. If she failed to properly determine the pairs of factors or miscalculated them, that would be her error. Without seeing her exact work, we can only generalize that the mistake lies in either identifying the correct factors or in misapplying algebraic principles while solving for \(x\).