To find the discriminant of the quadratic equation [tex]\(9x^2 + 2 = 10x\)[/tex], follow these steps:
1. Rewrite the equation in standard form:
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex].
Start by moving all terms to one side of the equation:
[tex]\[
9x^2 + 2 - 10x = 0
\][/tex]
Rearrange to get:
[tex]\[
9x^2 - 10x + 2 = 0
\][/tex]
2. Identify the coefficients:
From the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], identify [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
a = 9, \quad b = -10, \quad c = 2
\][/tex]
3. Write the formula for the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
4. Substitute the values into the discriminant formula:
[tex]\[
\Delta = (-10)^2 - 4 \cdot 9 \cdot 2
\][/tex]
5. Calculate the discriminant step by step:
- First, calculate [tex]\(b^2\)[/tex]:
[tex]\[
(-10)^2 = 100
\][/tex]
- Next, calculate [tex]\(4ac\)[/tex]:
[tex]\[
4 \cdot 9 \cdot 2 = 72
\][/tex]
- Finally, subtract [tex]\(4ac\)[/tex] from [tex]\(b^2\)[/tex]:
[tex]\[
\Delta = 100 - 72 = 28
\][/tex]
Thus, the discriminant of the quadratic equation [tex]\(9x^2 + 2 = 10x\)[/tex] is indeed [tex]\(28\)[/tex].
So, among the given options, the correct answer is:
[tex]\[ 28 \][/tex]
