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 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
No, Joana's conclusion is not correct. In Step 3, she made an error when factoring out the greatest common factor. The correct expression should be (2x+1)(2x-3)=0, which shows that the equation has two solutions: x = -1/2 and x = 3/2.
