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Solving the Equation: (x - 5)(x + 7) = 2x + 1
This article will walk you through the steps to solve the equation (x - 5)(x + 7) = 2x + 1.
1. Expand the Left-Hand Side
First, we need to expand the left-hand side of the equation by multiplying the two binomials:
(x - 5)(x + 7) = x² + 2x - 35
Now our equation becomes:
x² + 2x - 35 = 2x + 1
2. Move All Terms to One Side
To solve for x, we need to set the equation equal to zero. We can do this by subtracting 2x and 1 from both sides:
x² + 2x - 35 - 2x - 1 = 0
This simplifies to:
x² - 36 = 0
3. Solve the Quadratic Equation
The equation is now in the form of a quadratic equation (ax² + bx + c = 0). We can solve this using various methods, such as factoring or the quadratic formula.
In this case, the equation is easily factorable:
(x + 6)(x - 6) = 0
Now, we set each factor equal to zero and solve for x:
x + 6 = 0 or x - 6 = 0
Therefore, the solutions to the equation are:
x = -6 or x = 6
4. Verification
We can verify our solutions by plugging them back into the original equation:
For x = -6:
(-6 - 5)(-6 + 7) = 2(-6) + 1
(-11)(1) = -12 + 1
-11 = -11
This verifies that x = -6 is a valid solution.
For x = 6:
(6 - 5)(6 + 7) = 2(6) + 1
(1)(13) = 12 + 1
13 = 13
This verifies that x = 6 is also a valid solution.
In conclusion, the solutions to the equation (x - 5)(x + 7) = 2x + 1 are x = -6 and x = 6.