(4x-7)^2+16=40x-70

3 min read Jun 16, 2024

Solving the Equation (4x-7)^2 + 16 = 40x - 70

This article will guide you through the steps of solving the equation (4x-7)^2 + 16 = 40x - 70.

First, we need to expand the square on the left side of the equation:

(4x - 7)^2 = (4x - 7)(4x - 7) = 16x^2 - 56x + 49

Now the equation becomes:

16x^2 - 56x + 49 + 16 = 40x - 70

Combining the constants on the left side, we get:

16x^2 - 56x + 65 = 40x - 70

Next, move all terms to one side to set the equation to zero:

16x^2 - 96x + 135 = 0

The equation is now in the standard quadratic form (ax^2 + bx + c = 0). There are a few ways to solve this:

Factoring: Try to find two numbers that multiply to 135 and add up to -96. Unfortunately, this equation does not factor easily.
Quadratic Formula: The quadratic formula is a reliable method to solve any quadratic equation:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 16, b = -96, and c = 135. Substitute these values into the formula and simplify to find the solutions for x.

After applying the quadratic formula and simplifying, you will find two solutions for x. These are the values that make the original equation true.

Note: The solutions may be rational or irrational numbers. You may need to use a calculator to approximate the solutions.

By following these steps, you can successfully solve the equation (4x-7)^2 + 16 = 40x - 70 and find its solutions.