Question

Kevin is solving 6X squared - 7x + 2 = 0 by grouping he performed the first three steps as shown below however he struggles to complete step 4 cuz he could not come up with the shared if I don't know me on one step three what changes would get what should Kevin make to arrive at a standard binomial instep3 step 3 3x times 2x - 1 + 2 * -2x + 1 = 0

Answer 1

To solve the quadratic equation \(6x^2 - 7x + 2 = 0\) using the method of grouping, let’s begin by confirming Kevin's progress up to step 3.

The standard process involves rewriting the middle term using a combination of the coefficients. The equation can be factored by finding two numbers that multiply to \(6 \times 2 = 12\) and add to \(-7\). These numbers are \(-3\) and \(-4\).

Using these numbers, we can rewrite the equation:

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

Now, we can group the terms for factoring:

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

From here, we can factor out the common factors in each group:

1. From \(6x^2 - 3x\), factor out \(3x\):
\[
3x(2x - 1)
\]

2. From \(-4x + 2\), factor out \(-2\):
\[
-2(2x - 1)
\]

Putting it all together, we have:

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

Now, we can factor out the common binomial \(2x - 1\):

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

Finally, to solve for \(x\), we set each factor equal to zero:

1. \(2x - 1 = 0 \implies x = \frac{1}{2}\)
2. \(3x - 2 = 0 \implies x = \frac{2}{3}\)

In summary, Kevin's step 3 was almost correct. To arrive at the step 4 that forms standard binomials, Kevin needs to correctly factor out \(2x - 1\) and arrive at the expression \((2x - 1)(3x - 2) = 0\).