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.

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 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 .

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.

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.

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

Answer 1

Joana's conclusion is not correct. The issue she encountered in her work doesn't prevent the quadratic from having solutions.

When solving the equation \(4x^2 - 4x - 3 = 0\) by grouping, if the two groupings do not share a common binomial, she can still apply the quadratic formula or complete the square to find the solutions.

In fact, even if her method by grouping was unsuccessful, the quadratic equation can be solved using other methods. To determine the solutions of the quadratic equation, she could also apply the quadratic formula, which is given by:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \(a = 4\), \(b = -4\), and \(c = -3\).

Calculating the discriminant:

\[ b^2 - 4ac = (-4)^2 - 4(4)(-3) = 16 + 48 = 64 \]

Since the discriminant is positive (64), it indicates that there are two real solutions to the equation.

Thus, the correct answer is: No, because she can still continue the steps even if the two groupings have no shared binomial.