To find the discriminant of the quadratic equation [tex]\(2x + 5x^2 = 1\)[/tex], we first need to rewrite the equation in standard form. The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex].
Given equation:
[tex]\[ 2x + 5x^2 = 1 \][/tex]
Step 1: Rewrite the equation in standard form:
[tex]\[ 5x^2 + 2x - 1 = 0 \][/tex]
In this form:
- [tex]\(a\)[/tex] (the coefficient of [tex]\(x^2\)[/tex]) is 5,
- [tex]\(b\)[/tex] (the coefficient of [tex]\(x\)[/tex]) is 2,
- [tex]\(c\)[/tex] (the constant term) is -1.
Step 2: The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Step 3: Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ \Delta = (2)^2 - 4(5)(-1) \][/tex]
Calculating step-by-step:
[tex]\[ \Delta = 4 - (-20) \][/tex]
[tex]\[ \Delta = 4 + 20 \][/tex]
[tex]\[ \Delta = 24 \][/tex]
So, the discriminant of the given quadratic equation is 24.
