To solve the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex], we need to determine the discriminant, the number of roots, and then find the roots using the quadratic formula. Here's the step-by-step solution:
### Step 1: Determine the Discriminant
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are coefficients. For our equation, [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], and [tex]\(c = -4\)[/tex].
The discriminant ([tex]\(\Delta\)[/tex]) is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plug in the values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 7^2 - 4 \cdot 2 \cdot (-4)
\][/tex]
[tex]\[
\Delta = 49 + 32
\][/tex]
[tex]\[
\Delta = 81
\][/tex]
So, the discriminant is:
[tex]\[
\Delta = 81
\][/tex]
### Step 2: Determine the Number of Roots
The number of roots of a quadratic equation depends on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has 2 distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has 1 real root.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real roots (but has 2 complex roots).
Since [tex]\(\Delta = 81\)[/tex] (which is greater than 0), the equation has 2 distinct real roots.
### Step 3: Solve the Quadratic Equation Using the Quadratic Formula
The quadratic formula to find the roots is given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(\Delta = 81\)[/tex], [tex]\(a = 2\)[/tex], and [tex]\(b = 7\)[/tex] into the formula:
[tex]\[
x = \frac{-7 \pm \sqrt{81}}{2 \cdot 2}
\][/tex]
[tex]\[
x = \frac{-7 \pm 9}{4}
\][/tex]
Now, solve for the two roots:
Root 1:
[tex]\[
x_1 = \frac{-7 + 9}{4}
\][/tex]
[tex]\[
x_1 = \frac{2}{4}
\][/tex]
[tex]\[
x_1 = 0.5
\][/tex]
Root 2:
[tex]\[
x_2 = \frac{-7 - 9}{4}
\][/tex]
[tex]\[
x_2 = \frac{-16}{4}
\][/tex]
[tex]\[
x_2 = -4
\][/tex]
### Summary
The discriminant of the quadratic equation [tex]\(2x^2 + 7x - 4 = 0\)[/tex] is:
[tex]\[
\Delta = 81
\][/tex]
The number of roots is:
[tex]\[
\text{Number of roots} = 2
\][/tex]
The solutions to the quadratic equation are:
[tex]\[
x = 0.5 \quad \text{and} \quad x = -4
\][/tex]
So, the answers are:
a. [tex]\(\Delta = 81\)[/tex]
b. Number of roots = 2
c. Solutions: [tex]\(x = \frac{1}{2}\)[/tex] and [tex]\(x = -4\)[/tex]
