Question

-2x^2-3x+2=0 complete the table
x -3/4 substituted ? Evaluate? True ?
x -1/2. Substituted ? Evaluate? True?
x 1/2 substituted ? Evaluate? True ?
The only solution is x= ?

Answer 1

To solve the equation \(-2x^2 - 3x + 2 = 0\) and evaluate the given \(x\) values, I'll first find the solutions to the quadratic equation using the quadratic formula.

The quadratic formula is given by:

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

For the equation \(-2x^2 - 3x + 2 = 0\), the coefficients are:
- \(a = -2\)
- \(b = -3\)
- \(c = 2\)

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

\[
b^2 - 4ac = (-3)^2 - 4(-2)(2) = 9 + 16 = 25
\]

Now we calculate \(x\):

\[
x = \frac{{-(-3) \pm \sqrt{25}}}{2 \cdot (-2)} = \frac{{3 \pm 5}}{-4}
\]

Calculating the two possible values for \(x\):

1. When using the positive square root:
\[
x = \frac{{3 + 5}}{-4} = \frac{8}{-4} = -2
\]

2. When using the negative square root:
\[
x = \frac{{3 - 5}}{-4} = \frac{-2}{-4} = \frac{1}{2}
\]

Thus, the two solutions for the equation are \(x = -2\) and \(x = \frac{1}{2}\).

Now, I will evaluate the three specified \(x\) values in the equation:

### For \(x = -\frac{3}{4}\):
Substituting into the equation:

\[
-2\left(-\frac{3}{4}\right)^2 - 3\left(-\frac{3}{4}\right) + 2
\]
\[
= -2\left(\frac{9}{16}\right) + \frac{9}{4} + 2
\]
\[
= -\frac{18}{16} + \frac{36}{16} + \frac{32}{16}
\]
\[
= \frac{-18 + 36 + 32}{16} = \frac{50}{16} \neq 0 \text{ (False)}
\]

### For \(x = -\frac{1}{2}\):
Substituting into the equation:

\[
-2\left(-\frac{1}{2}\right)^2 - 3\left(-\frac{1}{2}\right) + 2
\]
\[
= -2\left(\frac{1}{4}\right) + \frac{3}{2} + 2
\]
\[
= -\frac{1}{2} + \frac{3}{2} + 2
\]
\[
= \frac{2}{2} + 2 = 1 + 2 = 3 \neq 0 \text{ (False)}
\]

### For \(x = \frac{1}{2}\):
Substituting into the equation:

\[
-2\left(\frac{1}{2}\right)^2 - 3\left(\frac{1}{2}\right) + 2
\]
\[
= -2\left(\frac{1}{4}\right) - \frac{3}{2} + 2
\]
\[
= -\frac{1}{2} - \frac{3}{2} + 2
\]
\[
= -\frac{1}{2} - \frac{3}{2} + \frac{4}{2} = 0 \text{ (True)}
\]

### Conclusion:
- The values you are checking yielded:
- For \(x = -\frac{3}{4}\): False
- For \(x = -\frac{1}{2}\): False
- For \(x = \frac{1}{2}\): True

The only solutions to the equation are \(x = -2\) and \(x = \frac{1}{2}\).