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 has no solutions. Is Joana’s conclusion correct? Review Joana’s work to answer the question.

Question

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  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

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

GPT-4o mini · Answer

Joana's conclusion is **not correct**. The reasoning is as follows: 1. **Step 1**: Joana rewrote the equation correctly to break down the middle term: $ 4x^2 + 2x - 6x - 3 = 0 $. 2. **Step 2**: She grouped the terms into two pairs: $ (4x^2 + 2x) + (-6x - 3) = 0 $. This is a valid step in the factoring process. 3. **Step 3**: In this step, she factored out $ 2x $ from the first group (resulting in $ 2x(2x + 1) $) and incorrectly factored out $ -3 $ from the second group, getting $ -3(2x + 1) $. While she noted that the two groupings did not share a common binomial, this alone does not mean the quadratic has no solutions. In fact, she can still continue the steps. Here's how she could proceed: If she rewrites Step 3 properly, she can put together the factors: $$ 2x(2x + 1) - 3(2x + 1) = 0 $$ This shows that $ (2x + 1)(2x - 3) = 0 $. Thus, if we solve this factored form, we find the solutions to be: $$ 2x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{2} $$ $$ 2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2} $$ Thus, there are indeed solutions to the quadratic equation $ 4x^2 - 4x - 3 = 0 $. Therefore, the correct answer is: **No, because she can still continue the steps even if the two groupings have no shared binomial.**