Certainly! Let's go step-by-step to solve this problem.
1. Problem Statement:
We are given that twice the square of a certain positive number, added to twice the number, equals 84. We need to find this number.
2. Formulating the Equation:
Let's denote the unknown number by [tex]\( x \)[/tex]. According to the problem:
- Twice the square of the number is represented as [tex]\( 2x^2 \)[/tex].
- Twice the number is represented as [tex]\( 2x \)[/tex].
- The sum of these two expressions is 84.
Therefore, we can write the equation:
[tex]\[
2x^2 + 2x = 84
\][/tex]
3. Simplifying the Equation:
To solve this quadratic equation, we should first simplify it. First, divide the whole equation by 2 to make it easier to manage:
[tex]\[
x^2 + x - 42 = 0
\][/tex]
4. Solving the Quadratic Equation:
We now solve the simplified quadratic equation [tex]\( x^2 + x - 42 = 0 \)[/tex]. To do this, we need to find the roots of the equation. This can be done using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our equation, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -42 \)[/tex].
Plugging these values into the quadratic formula:
[tex]\[
x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-42)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 168}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{169}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm 13}{2}
\][/tex]
5. Finding the Roots:
This gives us two possible solutions:
[tex]\[
x = \frac{-1 + 13}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x = \frac{-1 - 13}{2} = \frac{-14}{2} = -7
\][/tex]
6. Selecting the Positive Solution:
Since we are asked to find the positive number, we discard [tex]\( x = -7 \)[/tex]. Therefore, the positive solution is:
[tex]\[
x = 6
\][/tex]
Conclusion: The positive number that satisfies the given condition is [tex]\( \boxed{6} \)[/tex].
