Question

Joana solved the equation 4x2−4x−3=0%0D%0A4%0D%0A%0D%0A2%0D%0A−%0D%0A4%0D%0A%0D%0A−%0D%0A3%0D%0A=%0D%0A0%0D%0A . 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%0D%0A4%0D%0A%0D%0A2%0D%0A−%0D%0A4%0D%0A%0D%0A−%0D%0A3%0D%0A=%0D%0A0%0D%0A has no solutions. Is Joana’s conclusion correct? Review Joana’s work to answer the question. %0D%0A%0D%0AStep 1: 4x2+2x+(−6x)−3=0%0D%0A4%0D%0A%0D%0A2%0D%0A+%0D%0A2%0D%0A%0D%0A+%0D%0A(%0D%0A−%0D%0A6%0D%0A%0D%0A)%0D%0A−%0D%0A3%0D%0A=%0D%0A0%0D%0A %0D%0A%0D%0AStep 2: (4x2+2x)+(−6x−3)=0%0D%0A(%0D%0A4%0D%0A%0D%0A2%0D%0A+%0D%0A2%0D%0A%0D%0A)%0D%0A+%0D%0A(%0D%0A−%0D%0A6%0D%0A%0D%0A−%0D%0A3%0D%0A)%0D%0A=%0D%0A0%0D%0A %0D%0A%0D%0AStep 3: x(4x+2)+(−3(2x+1))=0

Answer 1

Let's analyze Joana’s work step by step to determine if her conclusion that the equation \( 4x^2 - 4x - 3 = 0 \) has no solutions is correct.

Starting with the quadratic equation:

### Step 1:

Joana rewrites the equation for grouping:

\[
4x^2 + 2x - 6x - 3 = 0
\]

This is a correct way to express \( -4x \) as \( 2x - 6x \).

### Step 2:

Next, she groups the terms:

\[
(4x^2 + 2x) + (-6x - 3) = 0
\]

This is also correct grouping of the terms.

### Step 3:

In her grouping, she attempts to factor:

\[
x(4x + 2) + (-3(2x + 1)) = 0
\]

Here, she started to factor out the common factors in each grouping. However, her groupings \( (4x + 2) \) and \( (-3(2x + 1)) \) do not share a common binomial, which complicates direct factorization.

Despite this, let's analyze whether the quadratic could have solutions using the quadratic formula instead.

### Quadratic Formula

The quadratic formula for \( ax^2 + bx + c = 0 \) is given by:

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

For the equation \( 4x^2 - 4x - 3 = 0 \):

- \( a = 4 \)
- \( b = -4 \)
- \( c = -3 \)

Calculating the discriminant \( b^2 - 4ac \):

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

The discriminant is \( 64 \), which is greater than 0, indicating that there are two distinct real solutions.

### Conclusion

Joana's conclusion that there are no solutions based on her observation of the grouping is incorrect. The quadratic equation \( 4x^2 - 4x - 3 = 0 \) does have solutions, confirmed by the positive discriminant. Thus, Joana's work in grouping did not lead to her conclusion correctly; instead, the quadratic formula indicates there are indeed solutions.