To determine if Joana's conclusion is correct, let's analyze her steps towards solving the quadratic equation \(4x^2 - 4x - 3 = 0\).
Step by step, Joana attempted to use the method of grouping by first rewriting the equation:
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Step 1: Joana rewrote the equation as \(4x^2 + 2x - 6x - 3 = 0\).
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Step 2: She grouped the terms as \((4x^2 + 2x) + (-6x - 3) = 0\).
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Step 3: In this step, she factored from the first grouping, obtaining \(x(4x + 2) - 3(2x + 1) = 0\).
At this point, Joana is trying to factor, but she incorrectly believes that the two groupings do not share a common binomial. In her expression, \(4x + 2\) and \(-3(2x + 1)\) do not share a common binomial factor, which is correct. However, she can still proceed with solving the equation without a common binomial factor.
Evaluating Joana's Conclusion:
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The fact that the groupings do not share a common binomial does not necessarily mean the original equation has no solutions. The equation could be factorable or solvable by other methods, like the quadratic formula.
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To check for solutions, we can also use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), given \(a = 4\), \(b = -4\), and \(c = -3\):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \] \[ = \frac{4 \pm \sqrt{16 + 48}}{8} \] \[ = \frac{4 \pm \sqrt{64}}{8} \] \[ = \frac{4 \pm 8}{8} \]
This yields two potential solutions:
- \(x = \frac{12}{8} = \frac{3}{2}\)
- \(x = \frac{-4}{8} = -\frac{1}{2}\)
Since this shows that the equation has solutions, Joana's conclusion that there are no solutions is incorrect.
Final Answer: No, because she can still continue the steps even if the two groupings have no shared binomial.
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