To solve this problem, let's break it down into clear, step-by-step calculations.
1. Identify the unknown variable:
Let the unknown number be [tex]\( x \)[/tex].
2. Set up the equation:
According to the problem statement, six times the sum of twice a number and 1 is equal to -6 times its square. This can be written as:
[tex]\[
6 \cdot (2x + 1) = -6 \cdot x^2
\][/tex]
3. Simplify the equation:
First, distribute the 6 on the left side:
[tex]\[
6 \cdot 2x + 6 \cdot 1 = -6 \cdot x^2
\][/tex]
This simplifies to:
[tex]\[
12x + 6 = -6x^2
\][/tex]
4. Rearrange the equation to standard quadratic form:
To do this, move all the terms to one side of the equation. Adding [tex]\( 6x^2 \)[/tex] to both sides, we get:
[tex]\[
6x^2 + 12x + 6 = 0
\][/tex]
5. Simplify the quadratic equation:
Notice that each term in the equation [tex]\( 6x^2 + 12x + 6 = 0 \)[/tex] can be divided by 6:
[tex]\[
x^2 + 2x + 1 = 0
\][/tex]
6. Factor the quadratic equation:
The quadratic equation [tex]\( x^2 + 2x + 1 \)[/tex] can be factored as:
[tex]\[
(x + 1)^2 = 0
\][/tex]
7. Solve for [tex]\( x \)[/tex]:
Setting the factored form to zero gives:
[tex]\[
x + 1 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -1
\][/tex]
Therefore, the number that satisfies the equation is [tex]\(-1\)[/tex].
