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:
- \(x = \frac{-23 + 47}{14} = \frac{24}{14} = \frac{12}{7}\)
- \(x = \frac{-23 - 47}{14} = \frac{-70}{14} = -5\)
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 \(x(x - 5)(x - 2)\):
- Expand \((x - 5)(x - 2)\):
\[ (x - 5)(x - 2) = x^2 - 2x - 5x + 10 = x^2 - 7x + 10 \]
- 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
- \(x = \frac{12}{7}\) and \(x = -5\)
- \(x = 1\), \(x = 3 + \sqrt{5}\), \(x = 3 - \sqrt{5}\)
- \(x = 3\)