Joana solved the equation 4x2−4x−3=0. She followed the steps to solve a quadratic equation by grouping. However, in Step 3 she noticed that the two groupings did not share a common binomial, as shown in her work below. She concluded that the quadratic equation has no solutions. Is Joana’s conclusion correct? Review Joana’s work to answer the question.

Question

Step 1: 4x2+2x+(−6x)−3=0
Step 2:(4x2+2x)+(−6x−3)=0
Step 3: x(4x+2x)+(−3(2x+1))=0

(1 point)
Responses

No, because she did not factor out the GCF from the first grouping.
No, because she did not factor out the GCF from the first grouping.

No, because she can still continue the steps even if the two groupings have no shared
binomial.
No, because she can still continue the steps even if the two groupings have no shared binomial.

Yes, because there are no factors of −12 that add up to −4.
Yes, because there are no factors of −12 that add up to −4.

Yes, because the two groupings do not share a common binomial

Answer 1

Joana's conclusion that the quadratic equation has no solutions is not correct.

The key point is that finding a common binomial in groupings is not strictly necessary to determine if a quadratic equation has solutions. Even though her grouping did not share a common binomial, the equation can still be factored or solved using other methods.

So, the correct response is:

No, because she can still continue the steps even if the two groupings have no shared binomial.