Let's analyze Joana’s work step by step to determine if her conclusion that the equation \( 4x^2 - 4x - 3 = 0 \) has no solutions is correct.
Starting with the quadratic equation:
Step 1:
Joana rewrites the equation for grouping:
\[ 4x^2 + 2x - 6x - 3 = 0 \]
This is a correct way to express \( -4x \) as \( 2x - 6x \).
Step 2:
Next, she groups the terms:
\[ (4x^2 + 2x) + (-6x - 3) = 0 \]
This is also correct grouping of the terms.
Step 3:
In her grouping, she attempts to factor:
\[ x(4x + 2) + (-3(2x + 1)) = 0 \]
Here, she started to factor out the common factors in each grouping. However, her groupings \( (4x + 2) \) and \( (-3(2x + 1)) \) do not share a common binomial, which complicates direct factorization.
Despite this, let's analyze whether the quadratic could have solutions using the quadratic formula instead.
Quadratic Formula
The quadratic formula for \( ax^2 + bx + c = 0 \) is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \( 4x^2 - 4x - 3 = 0 \):
- \( a = 4 \)
- \( b = -4 \)
- \( c = -3 \)
Calculating the discriminant \( b^2 - 4ac \):
\[ (-4)^2 - 4(4)(-3) = 16 + 48 = 64 \]
The discriminant is \( 64 \), which is greater than 0, indicating that there are two distinct real solutions.
Conclusion
Joana's conclusion that there are no solutions based on her observation of the grouping is incorrect. The quadratic equation \( 4x^2 - 4x - 3 = 0 \) does have solutions, confirmed by the positive discriminant. Thus, Joana's work in grouping did not lead to her conclusion correctly; instead, the quadratic formula indicates there are indeed solutions.
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