Answer :
To rewrite the equation [tex]\(x^2 + 20x + 98.5 = 0\)[/tex] in the form [tex]\((x-p)^2 = q\)[/tex], we will follow these steps:
1. Move the constant term to the other side: Start by isolating the quadratic and linear terms on one side.
[tex]\[
x^2 + 20x = -98.5
\][/tex]
2. Complete the square: We will add a number to both sides to make the left-hand side a perfect square trinomial. To do this, take half of the coefficient of [tex]\(x\)[/tex], square it, and add to both sides.
[tex]\[
\text{Coefficient of } x = 20 \quad \Rightarrow \quad \frac{20}{2} = 10
\][/tex]
[tex]\[
10^2 = 100
\][/tex]
Add 100 to both sides of the equation:
[tex]\[
x^2 + 20x + 100 = -98.5 + 100
\][/tex]
Simplifying the right side gives:
[tex]\[
x^2 + 20x + 100 = 1.5
\][/tex]
3. Factor the perfect square trinomial: The left-hand side is now a perfect square trinomial, which factors as:
[tex]\[
(x + 10)^2 = 1.5
\][/tex]
In conclusion, the equation rewritten in the form [tex]\((x-p)^2 = q\)[/tex] is:
[tex]\[
(x + 10)^2 = 1.5
\][/tex]
1. Move the constant term to the other side: Start by isolating the quadratic and linear terms on one side.
[tex]\[
x^2 + 20x = -98.5
\][/tex]
2. Complete the square: We will add a number to both sides to make the left-hand side a perfect square trinomial. To do this, take half of the coefficient of [tex]\(x\)[/tex], square it, and add to both sides.
[tex]\[
\text{Coefficient of } x = 20 \quad \Rightarrow \quad \frac{20}{2} = 10
\][/tex]
[tex]\[
10^2 = 100
\][/tex]
Add 100 to both sides of the equation:
[tex]\[
x^2 + 20x + 100 = -98.5 + 100
\][/tex]
Simplifying the right side gives:
[tex]\[
x^2 + 20x + 100 = 1.5
\][/tex]
3. Factor the perfect square trinomial: The left-hand side is now a perfect square trinomial, which factors as:
[tex]\[
(x + 10)^2 = 1.5
\][/tex]
In conclusion, the equation rewritten in the form [tex]\((x-p)^2 = q\)[/tex] is:
[tex]\[
(x + 10)^2 = 1.5
\][/tex]
