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 grouping 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.

Answer 1

To determine whether Joana's conclusion is correct, we first need to solve the given quadratic equation \(4x^2 - 4x - 3 = 0\) using the quadratic formula.

The general form of a quadratic equation is:

\[ ax^2 + bx + c = 0 \]

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

We can apply the quadratic formula:

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

First, we calculate the discriminant \(b^2 - 4ac\):

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

Since the discriminant is \(64\), which is positive, it indicates that there are two distinct real solutions. Now we can find the roots:

\[ x = \frac{{-(-4) \pm \sqrt{64}}}{{2 \cdot 4}} = \frac{{4 \pm 8}}{{8}} \]

Calculating the two solutions:

1. For \(x = \frac{{4 + 8}}{{8}} = \frac{12}{8} = \frac{3}{2}\)
2. For \(x = \frac{{4 - 8}}{{8}} = \frac{-4}{8} = -\frac{1}{2}\)

Thus, the solutions to the equation \(4x^2 - 4x - 3 = 0\) are \(x = \frac{3}{2}\) and \(x = -\frac{1}{2}\).

Conclusion: Joana's conclusion that the quadratic equation has no solutions is incorrect. The equation actually has two distinct real solutions.