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 groupings did not share a common binomial, as shown in her work below. She concluded that the quadratic equation 4x^2−4x−3=0

Question

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

(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 the two groupings do not share a common binomial.
Yes, because the two groupings do not share a common binomial.
Yes, because there are no factors of −12

that add up to −4.

Answer 1

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

While it is true that in her Step 3, the two groupings did not share a common binomial, it does not necessarily mean that the equation has no solutions. The method of grouping is just one way to look for solutions, and sometimes it may not result in a shared binomial, but that doesn’t imply the original equation has no solutions.

Instead, Joana could have continued with other methods to find the solutions, such as using the quadratic formula. The quadratic formula is applicable to any quadratic equation of the form \(ax^2 + bx + c = 0\) and will always provide the solutions.

Therefore, the correct answer is:

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