(2x+5)^2+(4x+10)(x-3)+x^2-6x+9=0

3 min read Jun 16, 2024

Solving the Quadratic Equation: (2x+5)^2+(4x+10)(x-3)+x^2-6x+9=0

This article will guide you through the process of solving the quadratic equation: (2x+5)^2+(4x+10)(x-3)+x^2-6x+9=0. We'll utilize algebraic manipulation and factorization techniques to find the solutions for x.

Simplifying the Equation

First, let's simplify the equation by expanding the terms:

(2x+5)^2: This expands to (2x+5)(2x+5) = 4x^2 + 20x + 25
(4x+10)(x-3): This expands to 4x^2 - 2x - 30
x^2-6x+9: This is already in its simplest form.

Now, let's substitute these expanded expressions back into the original equation:

4x^2 + 20x + 25 + 4x^2 - 2x - 30 + x^2 - 6x + 9 = 0

Combining Like Terms

Next, we combine the like terms to get a simplified quadratic equation:

9x^2 + 12x + 4 = 0

Factoring the Quadratic Equation

The simplified equation is now a standard quadratic equation in the form ax^2 + bx + c = 0. We can try to factor it to find the solutions.

In this case, we can notice that the equation is a perfect square trinomial: (3x + 2)^2 = 0

Finding the Solutions

Now, we can solve for x by taking the square root of both sides:

3x + 2 = 0

Solving for x, we get:

x = -2/3

Therefore, the solution to the quadratic equation (2x+5)^2+(4x+10)(x-3)+x^2-6x+9=0 is x = -2/3.

This solution is a double root, meaning it appears twice in the solution set. This signifies that the quadratic equation has only one distinct solution.