(1-3x^2)-(x-2)(9x+1)=(3x-4)(3x+4)-9(x+3)^2

4 min read Jun 16, 2024

Solving the Equation: (1-3x^2)-(x-2)(9x+1)=(3x-4)(3x+4)-9(x+3)^2

This article will guide you through the process of solving the given equation:

(1-3x^2)-(x-2)(9x+1)=(3x-4)(3x+4)-9(x+3)^2

The first step is to expand all the products in the equation. We'll use the distributive property (also known as FOIL) for this:

Now our equation looks like this:

(1-3x^2) - (9x^2 - 17x - 2) = (9x^2 - 16) - (9x^2 + 54x + 81)

Next, we need to simplify both sides of the equation by combining like terms:

Now our equation is:

-12x^2 + 17x + 3 = -54x - 97

To solve for x, we need to rearrange the equation so all terms are on one side and set it equal to zero:

-12x^2 + 17x + 3 + 54x + 97 = 0

-12x^2 + 71x + 100 = 0

We now have a quadratic equation in standard form. There are a few methods to solve for x:

Factoring: If the quadratic expression can be factored, we can find the values of x that make the expression equal to zero.
Quadratic Formula: This formula always provides the solutions to a quadratic equation:

x = [-b ± √(b^2 - 4ac)] / 2a

Where a, b, and c are the coefficients of the quadratic equation.

In this case, factoring might be difficult, so we'll use the quadratic formula.

Substituting these values into the quadratic formula:

x = [-71 ± √(71^2 - 4 * -12 * 100)] / (2 * -12)

x = [-71 ± √(9361 + 4800)] / -24

x = [-71 ± √(14161)] / -24

x = [-71 ± 119] / -24

This gives us two possible solutions:

x1 = (-71 + 119) / -24 = -2

x2 = (-71 - 119) / -24 = 8

Therefore, the solutions to the equation (1-3x^2)-(x-2)(9x+1)=(3x-4)(3x+4)-9(x+3)^2 are x = -2 and x = 8.