Question
Hyundai-eun is solving the quadratic equation x^2 - 4x - 12 = 0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error?
Answers
GPT-4o mini
To determine where Hyung-eun makes an error in solving the quadratic equation \(x^2 - 4x - 12 = 0\) by grouping, we first need to check each potential step in the process of solving a quadratic equation.
Typically, when using grouping to solve a quadratic equation, the approach often involves:
1. Writing the quadratic in standard form.
2. Finding two numbers that multiply to \(ac\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to \(b\) (the coefficient of \(x\)).
3. Rewriting the middle term using those two numbers to create a grouping.
4. Factoring by grouping.
1. The original equation is \(x^2 - 4x - 12 = 0\).
2. Here, \(a = 1\), \(b = -4\), and \(c = -12\), so we calculate \(ac = 1 \times -12 = -12\).
3. We need to find two numbers that multiply to \(-12\) and add to \(-4\). The correct numbers are \(-6\) and \(2\) since \(-6 \times 2 = -12\) and \(-6 + 2 = -4\).
4. Rewriting the equation using these numbers gives \(x^2 - 6x + 2x - 12 = 0\).
Next, we can group the terms:
\((x^2 - 6x) + (2x - 12) = 0\).
Now we factor out common factors from each group:
\(x(x - 6) + 2(x - 6) = 0\).
We can now factor by grouping:
\((x - 6)(x + 2) = 0\).
Now we can proceed to find the roots:
Setting each factor to zero gives:
\(x - 6 = 0 \Rightarrow x = 6\),
\(x + 2 = 0 \Rightarrow x = -2\).
In conclusion, if we see Hyung-eun's work divided correctly, there may be an error in the selection of factor pairs, a sign error in multiplication or addition, or in the grouping step. If you could provide specific steps that were attempted, I would be able to pinpoint the error in her calculations.
Typically, when using grouping to solve a quadratic equation, the approach often involves:
1. Writing the quadratic in standard form.
2. Finding two numbers that multiply to \(ac\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to \(b\) (the coefficient of \(x\)).
3. Rewriting the middle term using those two numbers to create a grouping.
4. Factoring by grouping.
1. The original equation is \(x^2 - 4x - 12 = 0\).
2. Here, \(a = 1\), \(b = -4\), and \(c = -12\), so we calculate \(ac = 1 \times -12 = -12\).
3. We need to find two numbers that multiply to \(-12\) and add to \(-4\). The correct numbers are \(-6\) and \(2\) since \(-6 \times 2 = -12\) and \(-6 + 2 = -4\).
4. Rewriting the equation using these numbers gives \(x^2 - 6x + 2x - 12 = 0\).
Next, we can group the terms:
\((x^2 - 6x) + (2x - 12) = 0\).
Now we factor out common factors from each group:
\(x(x - 6) + 2(x - 6) = 0\).
We can now factor by grouping:
\((x - 6)(x + 2) = 0\).
Now we can proceed to find the roots:
Setting each factor to zero gives:
\(x - 6 = 0 \Rightarrow x = 6\),
\(x + 2 = 0 \Rightarrow x = -2\).
In conclusion, if we see Hyung-eun's work divided correctly, there may be an error in the selection of factor pairs, a sign error in multiplication or addition, or in the grouping step. If you could provide specific steps that were attempted, I would be able to pinpoint the error in her calculations.