Question

7) Find the discriminant to determine the number of roots, then solve the quadratic equation using the quadratic formula.
2x^2+7x-4=0
____ discriminant
____ number of roots
____ solution

a=-81
b=2
c=1/2 and -4
d= -1/2 and 4
e= 81
f= 0
g= no solution
h=-2
i= 1

Answer 1

To solve the quadratic equation \(2x^2 + 7x - 4 = 0\), we first need to find the discriminant, which is given by the formula:

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

In this equation, \(a = 2\), \(b = 7\), and \(c = -4\).

Let's calculate the discriminant:

\[
D = 7^2 - 4 \cdot 2 \cdot (-4)
\]
\[
D = 49 + 32
\]
\[
D = 81
\]

### Discriminant:
- The discriminant \(D = 81\).

### Number of Roots:
The number of roots can be determined from the value of the discriminant:
- If \(D > 0\), there are 2 distinct real roots.
- If \(D = 0\), there is exactly 1 real root.
- If \(D < 0\), there are no real roots.

Since \(D = 81\) (which is greater than 0), there are **2 distinct real roots**.

### Solutions:
Now we will use the quadratic formula to find the roots of the equation:

\[
x = \frac{-b \pm \sqrt{D}}{2a}
\]

Using the values \(b = 7\), \(D = 81\), and \(a = 2\):

\[
x = \frac{-7 \pm \sqrt{81}}{2 \cdot 2}
\]
\[
x = \frac{-7 \pm 9}{4}
\]

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

1. For \(x = \frac{-7 + 9}{4}\):
\[
x = \frac{2}{4} = \frac{1}{2}
\]

2. For \(x = \frac{-7 - 9}{4}\):
\[
x = \frac{-16}{4} = -4
\]

### Solutions:
- The solutions are \(x = \frac{1}{2}\) and \(x = -4\).

### Summary:
- **Discriminant:** 81
- **Number of Roots:** 2 distinct real roots
- **Solutions:** \(x = \frac{1}{2}\) and \(x = -4\)