(x+7)^2=2x^2+16x+25

3 min read Jun 17, 2024

Solving the Equation: (x+7)^2 = 2x^2 + 16x + 25

This article will guide you through the steps to solve the equation (x+7)^2 = 2x^2 + 16x + 25. We'll explore the different methods and techniques involved in finding the solution(s) to this quadratic equation.

Expanding and Simplifying

Expand the left side of the equation: (x+7)^2 = (x+7)(x+7) = x^2 + 14x + 49
Rewrite the equation: x^2 + 14x + 49 = 2x^2 + 16x + 25
Rearrange the equation to the standard quadratic form (ax^2 + bx + c = 0): 0 = 2x^2 + 16x + 25 - x^2 - 14x - 49 0 = x^2 + 2x - 24

Solving the Quadratic Equation

Now we have a standard quadratic equation: x^2 + 2x - 24 = 0. There are several ways to solve it:

1. Factoring

Find two numbers that add up to 2 (coefficient of x) and multiply to -24 (constant term). In this case, the numbers are 6 and -4.
Factor the equation: (x + 6)(x - 4) = 0
Set each factor equal to zero and solve for x: x + 6 = 0 => x = -6 x - 4 = 0 => x = 4

2. Quadratic Formula

The quadratic formula is used to solve any quadratic equation in the form ax^2 + bx + c = 0: x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 1, b = 2, and c = -24. Substitute these values into the formula: x = (-2 ± √(2^2 - 4 * 1 * -24)) / (2 * 1) x = (-2 ± √(100)) / 2 x = (-2 ± 10) / 2
Solve for x: x = (-2 + 10) / 2 = 4 x = (-2 - 10) / 2 = -6

Conclusion

Therefore, the solutions to the equation (x+7)^2 = 2x^2 + 16x + 25 are x = 4 and x = -6. Both methods, factoring and the quadratic formula, lead to the same solutions.