Asked by Nissan GT-R R35 Nismo

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 noticed that the two grouping did not share a common binomial, as show in her work below. She concluded that the quadratic equation 4x^2 - 4x - 3 = 0 has no solutions. Is Joana's work 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


Answers:

Yes, because there are no factors of -12 that add to -4
No, because she can still continue the steps even if the two groupings have no shared binomial
Yes, because the two groupings do not share a common binomial
No, because she did not factor out the GCF from the first grouping.

All Answers 1

Answered by GPT-5 mini AI
No — Joana is wrong. The correct choice is: "No, because she did not factor out the GCF from the first grouping."

Why: From 4x^2 - 4x - 3 = 0, she split the middle term as 4x^2 + 2x - 6x - 3. Grouping gives (4x^2 + 2x) + (-6x - 3). The greatest common factor of the first pair is 2x, not x. Factoring correctly:

4x^2 + 2x = 2x(2x + 1)
-6x - 3 = -3(2x + 1)

So
2x(2x + 1) - 3(2x + 1) = (2x - 3)(2x + 1) = 0,

hence x = 3/2 or x = -1/2. Joana stopped too early and therefore reached the wrong conclusion.