Question

Consider the following equations. Solve them by showing your work clearly and step by step. Show a proof

1. 7x² + 23x = 60
2. x(x - 5)(x - 2) = 4
3. (x - 3)² = 0

Answer 1

Let's solve each equation step by step.

### 1. Solve the equation: \(7x^2 + 23x = 60\)

**Step 1: Rearrange the equation to standard quadratic form.**

Subtract 60 from both sides:

\[
7x^2 + 23x - 60 = 0
\]

**Step 2: Apply the quadratic formula.**

The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For our equation \(7x^2 + 23x - 60 = 0\), the coefficients are:
- \(a = 7\)
- \(b = 23\)
- \(c = -60\)

**Step 3: Calculate the discriminant (\(b^2 - 4ac\)).**

\[
b^2 - 4ac = 23^2 - 4 \cdot 7 \cdot (-60) = 529 + 1680 = 2209
\]

**Step 4: Calculate the roots.**

\[
x = \frac{-23 \pm \sqrt{2209}}{2 \cdot 7} = \frac{-23 \pm 47}{14}
\]

Now calculate two solutions:

1. \(x = \frac{-23 + 47}{14} = \frac{24}{14} = \frac{12}{7}\)
2. \(x = \frac{-23 - 47}{14} = \frac{-70}{14} = -5\)

**Final Solutions:**

\[
x = \frac{12}{7} \quad \text{and} \quad x = -5
\]

----

### 2. Solve the equation: \(x(x - 5)(x - 2) = 4\)

**Step 1: Expand the left side.**

First, expand \(x(x - 5)(x - 2)\):

1. Expand \((x - 5)(x - 2)\):

\[
(x - 5)(x - 2) = x^2 - 2x - 5x + 10 = x^2 - 7x + 10
\]

2. Multiply by \(x\):

\[
x(x^2 - 7x + 10) = x^3 - 7x^2 + 10x
\]

Now we rewrite the equation:

\[
x^3 - 7x^2 + 10x = 4
\]

**Step 2: Rearrange to standard form.**

Subtract 4 from both sides:

\[
x^3 - 7x^2 + 10x - 4 = 0
\]

**Step 3: Use the Rational Root Theorem.**

Let's test rational roots like \(x = 1\):

\[
1^3 - 7(1^2) + 10(1) - 4 = 1 - 7 + 10 - 4 = 0
\]

So, \(x = 1\) is a root. We can perform synthetic division to factor the cubic polynomial.

**Step 4: Factor out \((x - 1)\).**

Perform synthetic division of \(x^3 - 7x^2 + 10x - 4\) by \(x - 1\):

1 | 1 | -7 | 10 | -4
--|---|----|---|---
| | 1 | -6 | 4
| 1 | -6 | 4 | 0

The result is:

\[
x^2 - 6x + 4
\]

**Step 5: Solve \(x^2 - 6x + 4 = 0\) using the quadratic formula.**

\[
x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm \sqrt{20}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5}
\]

**Final Solutions:**

\[
x = 1, \quad x = 3 + \sqrt{5}, \quad x = 3 - \sqrt{5}
\]

----

### 3. Solve the equation: \((x - 3)^2 = 0\)

**Step 1: Take the square root.**

Taking the square root on both sides:

\[
x - 3 = 0
\]

**Step 2: Solve for \(x\).**

Add 3 to both sides:

\[
x = 3
\]

**Final Solution:**

\[
x = 3
\]

---

### Summary of Solutions

1. \(x = \frac{12}{7}\) and \(x = -5\)
2. \(x = 1\), \(x = 3 + \sqrt{5}\), \(x = 3 - \sqrt{5}\)
3. \(x = 3\)