Question

Use the discriminant to determine the number of real solutions of the equation. 4x2−3x +2=0 Part 1: Find the value of the discriminant. Show your work.

Answer 1

To determine the number of real solutions of the quadratic equation \(4x^2 - 3x + 2 = 0\), we first need to identify the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

For the given equation:

- \(a = 4\)
- \(b = -3\)
- \(c = 2\)

The discriminant \(D\) of a quadratic equation is given by the formula:

\[
D = b^2 - 4ac
\]

Now we can substitute the values of \(a\), \(b\), and \(c\) into the formula:

\[
D = (-3)^2 - 4(4)(2)
\]

Calculating \( (-3)^2 \):

\[
D = 9 - 4(4)(2)
\]

Calculating \( 4(4)(2) \):

\[
4(4)(2) = 32
\]

Now, substituting this back into the equation for \(D\):

\[
D = 9 - 32
\]

Computing \(9 - 32\):

\[
D = -23
\]

### Part 1: Value of the Discriminant
The value of the discriminant \(D\) is \(-23\).

### Part 2: Determine the Number of Real Solutions
The discriminant tells us about the nature of the roots:

- If \(D > 0\): Two distinct real solutions
- If \(D = 0\): One real solution (a repeated root)
- If \(D < 0\): No real solutions (the roots are complex)

Since the discriminant \(D = -23\) is less than 0, we can conclude that:

**There are no real solutions to the equation \(4x^2 - 3x + 2 = 0\).**