Sure, let's break down each expression step-by-step and solve or simplify them as needed.
### 1. Expression: [tex]\(8r - 11r^2 - 56\)[/tex]
1. Rewriting the expression:
[tex]\[
-11r^2 + 8r - 56
\][/tex]
2. Solving for [tex]\(r\)[/tex] when the expression is set to zero:
[tex]\[
-11r^2 + 8r - 56 = 0
\][/tex]
3. Using the quadratic formula [tex]\(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = -11\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = -56\)[/tex]:
[tex]\[
r = \frac{-8 \pm \sqrt{8^2 - 4(-11)(-56)}}{2(-11)}
\][/tex]
[tex]\[
r = \frac{-8 \pm \sqrt{64 - 2464}}{-22}
\][/tex]
[tex]\[
r = \frac{-8 \pm \sqrt{-2400}}{-22}
\][/tex]
[tex]\[
r = \frac{-8 \pm 10i\sqrt{6}}{-22}
\][/tex]
[tex]\[
r = \frac{4}{11} \pm \frac{10i\sqrt{6}}{11}
\][/tex]
Hence, the solutions are:
[tex]\[
r = \frac{4}{11} - \frac{10i\sqrt{6}}{11}, \quad r = \frac{4}{11} + \frac{10i\sqrt{6}}{11}
\][/tex]
### 2. Expression: [tex]\(92x - x^2 + 56\)[/tex]
1. Rewriting the expression:
[tex]\[
-x^2 + 92x + 56
\][/tex]
2. Solving for [tex]\(x\)[/tex] when the expression is set to zero:
[tex]\[
-x^2 + 92x + 56 = 0
\][/tex]
3. Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = -1\)[/tex], [tex]\(b = 92\)[/tex], and [tex]\(c = 56\)[/tex]:
[tex]\[
x = \frac{-92 \pm \sqrt{92^2 - 4(-1)(56)}}{2(-1)}
\][/tex]
[tex]\[
x = \frac{-92 \pm \sqrt{8464 + 224}}{-2}
\][/tex]
[tex]\[
x = \frac{-92 \pm \sqrt{8688}}{-2}
\][/tex]
[tex]\[
x = \frac{-92 \pm 2\sqrt{543}}{-2}
\][/tex]
[tex]\[
x = 46 \pm 2\sqrt{543}
\][/tex]
Hence, the solutions are:
[tex]\[
x = 46 - 2\sqrt{543}, \quad x = 46 + 2\sqrt{543}
\][/tex]
### 3. Expression: [tex]\(8(x^2 - 22x + 12x)\)[/tex]
1. Simplifying inside the parentheses:
[tex]\[
x^2 - 22x + 12x = x^2 - 10x
\][/tex]
2. Simplifying the whole expression:
[tex]\[
8(x^2 - 10x) = 8x^2 - 80x
\][/tex]
3. Solving for [tex]\(x\)[/tex] when the expression is set to zero:
[tex]\[
8x^2 - 80x = 0
\][/tex]
4. Factoring out 8x:
[tex]\[
8x(x - 10) = 0
\][/tex]
5. Setting each factor to zero:
[tex]\[
x = 0 \quad \text{or} \quad x - 10 = 0
\][/tex]
6. Solving for [tex]\(x\)[/tex]:
[tex]\[
x = 10
\][/tex]
Hence, the solutions are:
[tex]\[
x = 0, \quad x = 10
\][/tex]
### 4. Expression: [tex]\(0x^2 - 22n + 128\)[/tex]
1. Simplifying the expression:
[tex]\[
-22n + 128
\][/tex]
2. Solving for [tex]\(n\)[/tex] when the expression is set to zero:
[tex]\[
-22n + 128 = 0
\][/tex]
3. Isolating [tex]\(n\)[/tex]:
[tex]\[
-22n = -128
\][/tex]
[tex]\[
n = \frac{128}{22}
\][/tex]
[tex]\[
n = \frac{64}{11}
\][/tex]
Hence, the solution is:
[tex]\[
n = \frac{64}{11}
\][/tex]
This concludes the step-by-step solutions for each of the given expressions.
