(x-1)(x-2)%3D60.jpeg "(x+6)(x-1)(x-2)=60")
Solving the Cubic Equation: (x+6)(x-1)(x-2) = 60
This article will guide you through the steps to solve the cubic equation: (x+6)(x-1)(x-2) = 60.
1. Expanding the Equation
First, we need to expand the left side of the equation by multiplying the factors together.
- Step 1: Expand (x+6)(x-1) using the FOIL method: (x+6)(x-1) = x² + 5x - 6
- Step 2: Multiply the result by (x-2): (x² + 5x - 6)(x-2) = x³ + 3x² - 16x + 12
Now, the equation becomes: x³ + 3x² - 16x + 12 = 60
2. Transforming into a Standard Form
To solve the cubic equation, we need to bring all the terms to one side, resulting in a standard form:
x³ + 3x² - 16x - 48 = 0
3. Finding Solutions
Unfortunately, there's no simple formula to solve cubic equations directly like we have for quadratic equations. Here are a few methods you can try:
- Factoring: Try to factor the equation by grouping or using the Rational Root Theorem.
- Rational Root Theorem: This theorem helps identify potential rational roots. If you find a root, you can factor the equation further.
- Numerical Methods: You can use numerical methods like the Newton-Raphson method or graphing calculators to approximate the solutions.
4. Solution Example: Using Factoring
In this case, we can factor the equation by grouping:
- Step 1: Group the first two terms and the last two terms: (x³ + 3x²) + (-16x - 48) = 0
- Step 2: Factor out common factors: x²(x + 3) - 16(x + 3) = 0
- Step 3: Factor out (x + 3): (x + 3)(x² - 16) = 0
- Step 4: Factor the difference of squares: (x + 3)(x + 4)(x - 4) = 0
Therefore, the solutions are x = -3, x = -4, and x = 4.
Conclusion
Solving the cubic equation (x+6)(x-1)(x-2) = 60 involves expanding the equation, transforming it into standard form, and then applying methods like factoring or numerical techniques to find the solutions. In this particular example, factoring by grouping allowed us to find the solutions easily.