To determine whether Joana's conclusion is correct, let's first analyze the steps she took to solve the quadratic equation \(4x^2 - 4x - 3 = 0\) by grouping.
Step 1: Joana rewrote the equation as:
\[ 4x^{2} + 2x - 6x - 3 = 0. \]
She chose to break \(-4x\) into \(+2x - 6x\), which is valid.
Step 2: Joana grouped the terms:
\[ (4x^{2} + 2x) + (-6x - 3) = 0. \]
This is also correctly done.
Step 3: Joana factored out a common factor from each grouping:
\[ x(4x + 2) - 3(2x + 1) = 0. \]
At this point, Joana noticed that the two grouping factors \(4x + 2\) and \(2x + 1\) do not share a common binomial. This led her to conclude that the equation has no solutions.
Now, let's evaluate her conclusion based on the validity of her reasoning:
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Groupings not sharing a common binomial: While the two different groups in Step 3 do not have a common binomial factor, that does not invalidate the fact that the overall equation could still be solved.
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GCF from the first grouping: It is correct that she could have factored differently, but it's not the main issue. The goal is to find the solutions, and this step is just a different method of factoring.
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Finding solutions: The absence of a common binomial does not stop us from solving the quadratic equation. We can set each factor equal to zero for solutions.
Now, let's calculate the discriminant of the equation to find out if it has real solutions. The standard quadratic formula is:
\[ ax^2 + bx + c = 0, \] where \(a=4\), \(b=-4\), and \(c=-3\). The discriminant \(D\) is given by \(D = b^2 - 4ac\):
\[ D = (-4)^2 - 4(4)(-3) = 16 + 48 = 64. \]
Since the discriminant is positive (\(D > 0\)), this means there are two distinct real solutions for the equation.
Thus, Joana's conclusion that the quadratic equation \(4x^2 - 4x - 3 = 0\) has 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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