Answer :
To solve the problem, let's break it down into steps:
1. Understand the Problem Statement:
We need to find a number [tex]\( x \)[/tex] such that "four times the sum of my number and eleven is more than its square." This can be written as:
[tex]\[
4(x + 11) > x^2
\][/tex]
2. Set Up the Inequality:
Start by expanding the left side of the inequality:
[tex]\[
4(x + 11) = 4x + 44
\][/tex]
So, the inequality becomes:
[tex]\[
4x + 44 > x^2
\][/tex]
3. Rearrange the Inequality:
Move all terms to one side to set the inequality to zero:
[tex]\[
0 > x^2 - 4x - 44
\][/tex]
Alternatively, you can write it as:
[tex]\[
x^2 - 4x - 44 < 0
\][/tex]
4. Solve the Quadratic Inequality:
To solve this, we would typically find the roots of the quadratic equation [tex]\( x^2 - 4x - 44 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our case, [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -44 \)[/tex].
5. Find the Roots:
The discriminant is:
[tex]\[
b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-44) = 16 + 176 = 192
\][/tex]
The roots are:
[tex]\[
x = \frac{4 \pm \sqrt{192}}{2}
\][/tex]
6. Simplify the Roots:
Since [tex]\( \sqrt{192} = 8\sqrt{3} \)[/tex], the roots are:
[tex]\[
x = \frac{4 + 8\sqrt{3}}{2} = 2 + 4\sqrt{3}
\][/tex]
and
[tex]\[
x = \frac{4 - 8\sqrt{3}}{2} = 2 - 4\sqrt{3}
\][/tex]
7. Determine the Solution Interval:
The inequality [tex]\( x^2 - 4x - 44 < 0 \)[/tex] implies that [tex]\( x \)[/tex] must be between these two roots. Therefore, the solution is the interval:
[tex]\[
2 - 4\sqrt{3} < x < 2 + 4\sqrt{3}
\][/tex]
So, the number [tex]\( x \)[/tex] is within this range. This is the complete step-by-step solution to the problem. If you need further clarification or have more questions, feel free to ask!
1. Understand the Problem Statement:
We need to find a number [tex]\( x \)[/tex] such that "four times the sum of my number and eleven is more than its square." This can be written as:
[tex]\[
4(x + 11) > x^2
\][/tex]
2. Set Up the Inequality:
Start by expanding the left side of the inequality:
[tex]\[
4(x + 11) = 4x + 44
\][/tex]
So, the inequality becomes:
[tex]\[
4x + 44 > x^2
\][/tex]
3. Rearrange the Inequality:
Move all terms to one side to set the inequality to zero:
[tex]\[
0 > x^2 - 4x - 44
\][/tex]
Alternatively, you can write it as:
[tex]\[
x^2 - 4x - 44 < 0
\][/tex]
4. Solve the Quadratic Inequality:
To solve this, we would typically find the roots of the quadratic equation [tex]\( x^2 - 4x - 44 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
In our case, [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = -44 \)[/tex].
5. Find the Roots:
The discriminant is:
[tex]\[
b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-44) = 16 + 176 = 192
\][/tex]
The roots are:
[tex]\[
x = \frac{4 \pm \sqrt{192}}{2}
\][/tex]
6. Simplify the Roots:
Since [tex]\( \sqrt{192} = 8\sqrt{3} \)[/tex], the roots are:
[tex]\[
x = \frac{4 + 8\sqrt{3}}{2} = 2 + 4\sqrt{3}
\][/tex]
and
[tex]\[
x = \frac{4 - 8\sqrt{3}}{2} = 2 - 4\sqrt{3}
\][/tex]
7. Determine the Solution Interval:
The inequality [tex]\( x^2 - 4x - 44 < 0 \)[/tex] implies that [tex]\( x \)[/tex] must be between these two roots. Therefore, the solution is the interval:
[tex]\[
2 - 4\sqrt{3} < x < 2 + 4\sqrt{3}
\][/tex]
So, the number [tex]\( x \)[/tex] is within this range. This is the complete step-by-step solution to the problem. If you need further clarification or have more questions, feel free to ask!
