Question

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

Answer 1

-x² +4x +3 = x² - x - (1+2x²)

x² - x - 1 -2x² + x² - 4x - 3 = 0

- 5х - 4 = 0

-5х = 4

х = - 4/5

Answer 2

To solve the equation, we need to find the value(s) of x that satisfy it. Let's first combine like terms on both sides of the equation:

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

To eliminate the parentheses on the right side, distribute the negative sign:

-x^2 + 4x + 3 = x^2 - x - 1 - 2x^2

Next, combine the x^2 terms on one side and the constant terms on the other side:

-x^2 + x^2 + 4x + x + 3 + 1 = -2x^2

Simplify:

5x + 4 = -2x^2

Now, let's move all the terms to one side to create a quadratic equation in standard form (ax^2 + bx + c = 0):

2x^2 + 5x + 4 = 0

Now, we can attempt to factor or use the quadratic formula to find the solutions for x. However, it appears that the quadratic equation does not factor easily. Let's proceed with the quadratic formula:

The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

In our case, a = 2, b = 5, and c = 4. Substituting these values into the quadratic formula:

x = (-(5) ± √(5^2 - 4 * 2 * 4)) / 2 * 2 x = (-5 ± √(25 - 32)) / 4 x = (-5 ± √(-7)) / 4

Since the discriminant (the value inside the square root) is negative, there are no real solutions for x in this case. The equation has two complex solutions:

x = (-5 + √7i) / 4 x = (-5 - √7i) / 4

Where 'i' is the imaginary unit, i.e., √(-1).

So, the solutions to the original equation are complex numbers, specifically:

x = (-5 + √7i) / 4 x = (-5 - √7i) / 4