Which statement about the following equation is true?

[tex]\[2x^2 - 9x + 2 = -1\][/tex]

A. The discriminant is less than 0, so there are two real roots.
B. The discriminant is less than 0, so there are two complex roots.
C. The discriminant is greater than 0, so there are two real roots.
D. The discriminant is greater than 0, so there are two complex roots.



Answer :

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.