Joana solved the equation 4x^2 - 4x - 3 = 0. She followed the steps to solve a quadratic equation by grouping. however, in Step 3 she notices that the two groupings did not share a common binomial, as shown in her work below. She concluded that the quadratic equation 4x^2 - 4x - 3 = 0 has no solutions. Is Joana's conclusion correct? Review Joana's work to answer the question.

Question

Joana solved the equation 4x^2 - 4x - 3 = 0. She followed the steps to solve a quadratic equation by grouping. however, in Step 3 she notices that the two groupings did not share a common binomial, as shown in her work below. She concluded that the quadratic equation 4x^2 - 4x - 3 = 0 has no solutions. Is Joana's conclusion correct? Review Joana's work to answer the question.

Step 1: 4x^2 + 2x + (-6x) - 3 = 0
Step 2: (4x^2 + 2x) + (-6x - 3) = 0
Step 3: x (4x + 2) + (-3 (2x + 1)) = 0
A. Yes, because there is no factors of -12 that add up to -4.
B. Yes, because the two groupings do not share a common binomial.
C. No, because she did not factor out the GCF from the first grouping.
D. No, because she can still continue the steps even if the two groupings have no shared binomial.

Answer 1

Joana's conclusion is incorrect. The fact that the two groupings do not share a common binomial does not necessarily mean that the quadratic equation has no solutions. Joana can continue solving the equation by factoring out the GCF from each grouping and then using the zero-product property to find the solutions. Therefore, the correct answer is D. No, because she can still continue the steps even if the two groupings have no shared binomial.