$$ \left. \begin{array} { l } { 8 ( x ^ { 3 } + 80 x ^ { 2 } + 6 x = 0 } \ { ( x + 2 ) ( x - 3 ) = 50 } \ { ( x - 5 ) ( x - 1 ) = 3 } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 8 ( x ^ { 3 } + 80 x ^ { 2 } + 6 x = 0 } \ { ( x + 2 ) ( x - 3 ) = 50 } \ { ( x - 5 ) ( x - 1 ) = 3 } \end{array} ight. $$

UpStudy AI · Accepted Answer

Certainly! Let's solve each of the equations you've provided separately.

You have the following system of equations:

$$
\begin{cases}
8x^3 + 80x^2 + 6x = 0 \
(x + 2)(x - 3) = 50 \
(x - 5)(x - 1) = 3 \
\end{cases}
$$

### 1. **Solving $$ 8x^3 + 80x^2 + 6x = 0 $$**

First, factor out the common term $$ x $$:

$$
x(8x^2 + 80x + 6) = 0
$$

This gives us one real solution immediately:

$$
x = 0
$$

Now, solve the quadratic equation $$ 8x^2 + 80x + 6 = 0 $$:

$$
x = \frac{-80 \pm \sqrt{80^2 - 4 \cdot 8 \cdot 6}}{2 \cdot 8} = \frac{-80 \pm \sqrt{6400 - 192}}{16} = \frac{-80 \pm \sqrt{6208}}{16}
$$

Simplify the square root:

$$
\sqrt{6208} \approx 78.78
$$

So, the approximate solutions are:

$$
x \approx \frac{-80 \pm 78.78}{16}
$$

Which gives:

$$
x \approx \frac{-80 + 78.78}{16} \approx -0.07 \quad \text{and} \quad x \approx \frac{-80 - 78.78}{16} \approx -9.90
$$

**Solutions for the first equation:**

$$
x = 0,\ x \approx -0.07,\ \text{and}\ x \approx -9.90
$$

### 2. **Solving $$ (x + 2)(x - 3) = 50 $$**

First, expand the left side:

$$
x^2 - 3x + 2x - 6 = x^2 - x - 6 = 50
$$

Rearrange the equation:

$$
x^2 - x - 56 = 0
$$

Now, solve the quadratic equation:

$$
x = \frac{1 \pm \sqrt{1 + 224}}{2} = \frac{1 \pm \sqrt{225}}{2} = \frac{1 \pm 15}{2}
$$

This yields two solutions:

$$
x = \frac{1 + 15}{2} = 8 \quad \text{and} \quad x = \frac{1 - 15}{2} = -7
$$

**Solutions for the second equation:**

$$
x = 8\quad \text{and}\quad x = -7
$$

### 3. **Solving $$ (x - 5)(x - 1) = 3 $$**

First, expand the left side:

$$
x^2 - x - 5x + 5 = x^2 - 6x + 5 = 3
$$

Rearrange the equation:

$$
x^2 - 6x + 2 = 0
$$

Now, solve the quadratic equation:

$$
x = \frac{6 \pm \sqrt{36 - 8}}{2} = \frac{6 \pm \sqrt{28}}{2} = \frac{6 \pm 2\sqrt{7}}{2} = 3 \pm \sqrt{7}
$$

**Solutions for the third equation:**

$$
x = 3 + \sqrt{7} \approx 5.65\quad \text{and}\quad x = 3 - \sqrt{7} \approx 0.35
$$

### **Summary of All Solutions**

1. **First Equation ($$ 8x^3 + 80x^2 + 6x = 0 $$)**:
   - $$ x = 0 $$
   - $$ x \approx -0.07 $$
   - $$ x \approx -9.90 $$

2. **Second Equation ($$ (x + 2)(x - 3) = 50 $$)**:
   - $$ x = 8 $$
   - $$ x = -7 $$

3. **Third Equation ($$ (x - 5)(x - 1) = 3 $$)**:
   - $$ x \approx 5.65 $$
   - $$ x \approx 0.35 $$

Each equation is independent, and the solutions provided are specific to each individual equation. If you intended to treat these as a system of equations to be solved simultaneously, please note that there is no common solution that satisfies all three equations at the same time.

Feel free to ask if you need further clarification or assistance!
The solutions are: - First equation: $$ x = 0 $$, $$ x \approx -0.07 $$, $$ x \approx -9.90 $$ - Second equation: $$ x = 8 $$, $$ x = -7 $$ - Third equation: $$ x \approx 5.65 $$, $$ x \approx 0.35 $$ No common solution exists for all three equations simultaneously.

\( \left. \begin{array} { l } { 8 ( x ^ { 3 } + 80 x ^ { 2 } + 6 x = 0 } \\ { ( x + 2 ) ( x - 3 ) = 50 } \\ { ( x - 5 ) ( x - 1 ) = 3 } \end{array} \right. \)

Real Tutor Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions