Question

5(x^2 - 3x) + 2(4x - 7) = 3(2x + 1)^2 - 5x

Answer 1

To solve the equation 5(x^2 - 3x) + 2(4x - 7) = 3(2x + 1)^2 - 5x, we can follow these steps:

1. Simplify both sides of the equation:

Distribute and simplify:

5x^2 - 15x + 8x - 14 = 12x^2 + 12x + 3 - 5x

Simplify further:

5x^2 - 7x - 14 = 12x^2 + 7x + 3

2. Move all terms to one side of the equation:

Subtract 5x^2 and 7x from both sides:

0 = 7x^2 + 14x + 3

3. Set the equation equal to zero:

Rearrange the terms:

7x^2 + 14x + 3 = 0

4. Solve the quadratic equation:

Factoring this quadratic equation can be a bit complex. Instead, we can use the quadratic formula to find the solutions. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation 7x^2 + 14x + 3 = 0, we have:

a = 7, b = 14, c = 3

Substituting these values into the quadratic formula:

x = (-14 ± √(14^2 - 4 * 7 * 3)) / (2 * 7)

Simplifying:

x = (-14 ± √(196 - 84)) / 14

x = (-14 ± √112) / 14

x = (-14 ± 4√7) / 14

Simplifying further:

x = (-7 ± 2√7) / 7

x = -1 ± (2/√7)

Therefore, the solutions to the equation 5(x^2 - 3x) + 2(4x - 7) = 3(2x + 1)^2 - 5x are:

x = -1 + (2/√7) and x = -1 - (2/√7)

CROWN CROWN CROWN CROWN

Answer 2

To solve the equation 5(x^2 - 3x) + 2(4x - 7) = 3(2x + 1)^2 - 5x, follow these steps:

Expand and simplify both sides of the equation.

On the left side, expand the expressions within the parentheses:

5x^2 - 15x + 8x - 14 = 3(4x^2 + 4x + 1) - 5x

Now, simplify further:

5x^2 - 7x - 14 = 12x^2 + 12x + 3 - 5x

Combine like terms on both sides of the equation:

5x^2 - 7x - 14 = 12x^2 + 12x + 3 - 5x

First, subtract the terms on the right side from the left side:

5x^2 - 7x - 14 - 12x^2 - 12x - 3 + 5x = 0

Combine like terms on the left side:

(5x^2 - 12x^2) + (-7x - 12x + 5x) - 14 - 3 = 0

-7x^2 - 14x - 17 = 0

Now, we have a quadratic equation. To solve for x, you can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In this equation, a = -7, b = -14, and c = -17. Plug these values into the quadratic formula:

x = (-(-14) ± √((-14)² - 4(-7)(-17))) / (2(-7))

x = (14 ± √(196 - 476)) / (-14)

x = (14 ± √(-280)) / (-14)

Since the discriminant (the value inside the square root) is negative, this equation has complex solutions:

x = (14 ± √280i) / (-14)

Simplified:

x = -1 ± √5i

So, the solutions for x are x = -1 + √5i and x = -1 - √5i.