(3x%2B5)%3D-14.jpeg "(2x-3)(3x+5)=-14")
Solving the Equation: (2x-3)(3x+5) = -14
This article will guide you through the process of solving the equation (2x-3)(3x+5) = -14.
Expanding the Equation
First, we need to expand the left side of the equation by multiplying the two binomials:
(2x-3)(3x+5) = 6x² + 10x - 9x - 15
Simplifying the expression:
6x² + x - 15 = -14
Transforming into a Quadratic Equation
Now, move the constant term from the right side to the left side:
6x² + x - 15 + 14 = 0
This simplifies to:
6x² + x - 1 = 0
Solving the Quadratic Equation
We now have a standard quadratic equation in the form ax² + bx + c = 0. There are several ways to solve this, including:
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Factoring: Try to find two numbers that multiply to give ac (6 * -1 = -6) and add up to b (1). In this case, the numbers are 3 and -2. We can then rewrite the equation as:
6x² + 3x - 2x - 1 = 0
Factoring by grouping:
3x(2x + 1) - 1(2x + 1) = 0
(3x - 1)(2x + 1) = 0
Therefore, the solutions are:
3x - 1 = 0 or 2x + 1 = 0
x = 1/3 or x = -1/2
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Quadratic Formula: The quadratic formula is a general solution for any quadratic equation:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 6, b = 1, and c = -1. Plugging these values into the formula:
x = (-1 ± √(1² - 4 * 6 * -1)) / (2 * 6)
x = (-1 ± √(25)) / 12
x = (-1 ± 5) / 12
Therefore, the solutions are:
x = 1/3 or x = -1/2
Conclusion
The solutions to the equation (2x-3)(3x+5) = -14 are x = 1/3 and x = -1/2. Both methods, factoring and the quadratic formula, lead to the same answer.