(x%2B6)%3D2(3x%2B4).jpeg "(x+5)(x+6)=2(3x+4)")
Solving the Equation (x+5)(x+6) = 2(3x+4)
This article will guide you through the steps of solving the equation (x+5)(x+6) = 2(3x+4).
1. Expand Both Sides of the Equation
First, we need to expand both sides of the equation to get rid of the parentheses.
- Left side: (x+5)(x+6) = x² + 6x + 5x + 30 = x² + 11x + 30
- Right side: 2(3x+4) = 6x + 8
Now our equation looks like this: x² + 11x + 30 = 6x + 8
2. Rearrange the Equation
To solve for x, we need to have all the terms on one side of the equation. We can achieve this by subtracting 6x and 8 from both sides:
x² + 11x + 30 - 6x - 8 = 0
This simplifies to: x² + 5x + 22 = 0
3. Solve the Quadratic Equation
We now have a quadratic equation in the form ax² + bx + c = 0. There are a couple of ways to solve this:
-
Factoring: In this case, the equation doesn't factor easily.
-
Quadratic Formula: The quadratic formula is a general solution to all quadratic equations. It states that:
x = [-b ± √(b² - 4ac)] / 2a
Where:
- a = 1
- b = 5
- c = 22
Plugging these values into the formula:
x = [-5 ± √(5² - 4 * 1 * 22)] / 2 * 1 x = [-5 ± √(-63)] / 2 x = [-5 ± √63 * i] / 2 (where 'i' is the imaginary unit, √-1)
Therefore, the solutions to the equation are:
x = (-5 + √63 * i) / 2 and x = (-5 - √63 * i) / 2
Conclusion
The equation (x+5)(x+6) = 2(3x+4) has two solutions, both of which are complex numbers: x = (-5 + √63 * i) / 2 and x = (-5 - √63 * i) / 2.