(x-5)(x+7)=2x+1

3 min read Jun 17, 2024
(x-5)(x+7)=2x+1

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.

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