(2x+3)^2-4(2x+3)-12=0

2 min read Jun 16, 2024

Solving the Quadratic Equation: (2x+3)^2 - 4(2x+3) - 12 = 0

This article will guide you through solving the quadratic equation (2x+3)^2 - 4(2x+3) - 12 = 0. We'll use a combination of factoring and the quadratic formula to find the solutions.

Recognizing the Pattern

First, we can observe that the equation has a repeated expression: (2x+3). This suggests a potential simplification through substitution.

Substitution

Let's substitute 'y' for (2x+3):

y = (2x+3)

Now our equation becomes:

y^2 - 4y - 12 = 0

Factoring the Quadratic

This quadratic expression is now easier to factor. We need to find two numbers that multiply to -12 and add up to -4. These numbers are -6 and 2.

Therefore, we can factor the equation as follows:

(y - 6)(y + 2) = 0

Solving for y

For the product of two factors to be zero, at least one of them must be zero. So we have two possible solutions:

y - 6 = 0 => y = 6
y + 2 = 0 => y = -2

Back Substitution

Now, we need to substitute back the original expression for 'y':

2x + 3 = 6
2x + 3 = -2

Solving for x

Solving for 'x' in each equation:

2x = 3 => x = 3/2
2x = -5 => x = -5/2

Conclusion

Therefore, the solutions to the quadratic equation (2x+3)^2 - 4(2x+3) - 12 = 0 are:

x = 3/2 and x = -5/2