To answer the questions related to the quadratic equation [tex]\(9x^2 - 6x - 4 = -5\)[/tex], let's go through the process step by step.
Step 1: Simplify the equation
First, we need to move all terms to one side of the equation to set it to 0. So, starting with:
[tex]\[ 9x^2 - 6x - 4 = -5 \][/tex]
Add 5 to both sides:
[tex]\[ 9x^2 - 6x - 4 + 5 = 0 \][/tex]
This simplifies to:
[tex]\[ 9x^2 - 6x + 1 = 0 \][/tex]
Now, we have the quadratic equation in the form [tex]\(ax^2 + bx + c = 0\)[/tex] with:
[tex]\[ a = 9 \][/tex]
[tex]\[ b = -6 \][/tex]
[tex]\[ c = 1 \][/tex]
Step 2: Calculate the discriminant
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the given values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-6)^2 - 4 \cdot 9 \cdot 1 \][/tex]
[tex]\[ \Delta = 36 - 36 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
Thus, the value of the discriminant is [tex]\(0\)[/tex].
Step 3: Determine the number of roots
The number of roots of a quadratic equation depends on the value of the discriminant ([tex]\(\Delta\)[/tex]):
- If [tex]\(\Delta > 0\)[/tex], there are 2 distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], there is 1 real root (or a repeated root).
- If [tex]\(\Delta < 0\)[/tex], there are no real roots (but 2 complex roots).
Since the discriminant [tex]\(\Delta\)[/tex] for our equation is 0, it means there is exactly 1 real root.
Summary:
1. The value of the discriminant is [tex]\(0\)[/tex].
2. There is 1 real root of the equation.
