10.

Question

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 to 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

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

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 conyinue the steps even if the two groupings have no shared binomial.

Answer 1

Joana's conclusion is not correct. The correct response is:

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

Just because the factorizations don't share a common binomial does not mean that the quadratic has no solutions. In fact, she can still use other methods to solve the equation, such as the quadratic formula or completing the square. The equation \(4x^2 - 4x - 3 = 0\) can actually be solved using those methods. Thus, concluding that there are no solutions based on her grouping method is incorrect.