Joana solved the equation 4x2−4x−3=0

Question

. 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 4x2−4x−3=0
has no solutions. Is Joana’s conclusion correct? Review Joana’s work to answer the question.

Step 1: 4x2+2x+(−6x)−3=0


Step 2: (4x2+2x)+(−6x−3)=0


Step 3: x(4x+2)+(−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 negative 12 that add up to negative 4 .

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

Answer 1

Joana's conclusion that the quadratic equation \( 4x^2 - 4x - 3 = 0 \) has no solutions is not correct.

The appropriate response is:

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

In the process of factoring quadratic equations by grouping, it is possible to proceed with the solution even if the two groupings do not share a common factor. In this case, after identifying the groupings and factoring them separately, Joana could set each factor equal to zero to find the possible solutions for \( x \). So the conclusion that the equation has no solutions based on the lack of a common binomial is incorrect.

Additionally, the equation \( 4x^2 - 4x - 3 \) can also be solved using the quadratic formula, providing further evidence that solutions do exist.