Joana's conclusion is not correct. The fact that she did not find a common binomial factor in her grouping does not mean that the quadratic equation has no solutions.
Here's why:
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Reviewing her steps:
- In Step 1, she correctly rewrote the middle term \( -4x \) as \( 2x - 6x \).
- In Step 2, she grouped the terms as \( (4x^2 + 2x) + (-6x - 3) = 0 \).
- However, in Step 3, her factoring is not accurate. The correct grouping and factoring should lead to finding common factors.
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Finding solutions:
- If the grouping fails to yield a common binomial factor, it doesn't imply no solutions exist; it merely indicates that this method of factoring might not work. The quadratic formula could still provide solutions, regardless of whether she was able to factor it through grouping.
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Using the Quadratic Formula:
- The equation \( 4x^2 - 4x - 3 = 0 \) can be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For this equation:
- \( a = 4 \)
- \( b = -4 \)
- \( c = -3 \)
- The discriminant \( b^2 - 4ac = (-4)^2 - 4(4)(-3) = 16 + 48 = 64 \), which is positive, indicating that there are two real solutions.
- The equation \( 4x^2 - 4x - 3 = 0 \) can be solved using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For this equation:
Therefore, the correct response is: No, because she can still continue the steps even if the two groupings have no shared binomial.
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