Joana's conclusion that the quadratic equation \(4x^2 - 4x - 3 = 0\) has no solutions is incorrect. The error lies in her attempt to factor the equation by grouping and her reasoning based on the lack of a common binomial.
Let's break down Joana's work:
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In Step 1: \(4x^2 + 2x - 6x - 3 = 0\) is correctly rewritten as \(4x^2 - 4x - 3 = 0\).
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In Step 2: She grouped the terms into \((4x^2 + 2x) + (-6x - 3) = 0\).
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In Step 3, she factored out \(x\) from the first group and \(-3\) from the second group, which was not done correctly. A common factor needs to be factored out correctly to identify any shared binomials.
However, Joana's reasoning about concluding that the equation has no solutions based on the lack of common binomials is flawed. Even if the grouping does not lead to a common factor, it is still possible to solve the quadratic equation using the quadratic formula or by completing the square.
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \(4x^2 - 4x - 3 = 0\):
- Here, \(a = 4\), \(b = -4\), and \(c = -3\).
- The discriminant (\(b^2 - 4ac\)) is:
\[ (-4)^2 - 4(4)(-3) = 16 + 48 = 64 \]
Since the discriminant is positive, it indicates that there are two distinct real solutions.
Thus, based on the evaluation of her method and the use of the quadratic formula, the correct response to Joana’s conclusion is:
No, because she can still continue the steps even if the two groupings have no shared binomial.
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