To determine which statement about the given quadratic equation [tex]\(2x^2 - 9x + 2 = -1\)[/tex] is true, let’s follow the steps of solving it and analyzing its roots.
1. Simplify the Equation:
First, add 1 to both sides of the equation to bring it to the standard form of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
2x^2 - 9x + 2 + 1 = 0
\][/tex]
Simplify the constant terms:
[tex]\[
2x^2 - 9x + 3 = 0
\][/tex]
2. Identify the Coefficients:
The equation is now in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex], with:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -9\)[/tex]
- [tex]\(c = 3\)[/tex]
3. Calculate 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]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-9)^2 - 4 \cdot 2 \cdot 3
\][/tex]
Calculate the discriminant:
[tex]\[
\Delta = 81 - 24 = 57
\][/tex]
4. Analyze the Discriminant:
The value of the discriminant [tex]\(\Delta\)[/tex] is 57.
- If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the quadratic equation has exactly one real root (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], the quadratic equation has two complex (conjugate) roots.
Since [tex]\(\Delta = 57 > 0\)[/tex], the quadratic equation [tex]\(2x^2 - 9x + 3 = 0\)[/tex] has two distinct real roots.
Therefore, the correct statement about the equation [tex]\(2x^2 - 9x + 2 = -1\)[/tex] is:
The discriminant is greater than 0, so there are two real roots.
