Sure, let's solve the equation and find the quadratic, linear, and constant terms along with the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
Given equation:
[tex]\[
(2x + 5)(x - 1) + 6 = 0
\][/tex]
First, we need to expand the product [tex]\((2x + 5)(x - 1)\)[/tex].
1. Expand the product:
[tex]\[
(2x + 5)(x - 1) = 2x \cdot x + 2x \cdot (-1) + 5 \cdot x + 5 \cdot (-1)
\][/tex]
Calculate each term:
[tex]\[
= 2x^2 - 2x + 5x - 5
\][/tex]
Combine like terms:
[tex]\[
= 2x^2 + 3x - 5
\][/tex]
2. Add the constant 6 to this result:
[tex]\[
2x^2 + 3x - 5 + 6
\][/tex]
3. Simplify the equation:
[tex]\[
2x^2 + 3x + 1 = 0
\][/tex]
So the expanded equation is:
[tex]\[
2x^2 + 3x + 1 = 0
\][/tex]
From this equation, we can identify the coefficients of the terms:
- The quadratic term [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex]:
[tex]\[
a = 2
\][/tex]
- The linear term [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]:
[tex]\[
b = 3
\][/tex]
- The constant term [tex]\(c\)[/tex]:
[tex]\[
c = 1
\][/tex]
Therefore, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
a = 2, \quad b = 3, \quad c = 1
\][/tex]
These coefficients tell us that the quadratic function in standard form is [tex]\(2x^2 + 3x + 1\)[/tex].
