Question

Решить уравнение |- x^2 + 2x -3| = 1

Answer 1

|-3|=1
3=1
Выражение ложно

Answer 2

To solve the equation |-x^2 + 2x - 3| = 1, we need to consider both the positive and negative values of the expression inside the absolute value separately.

1. Positive Expression: Solve the equation inside the absolute value for when it is positive, and then set it equal to 1:

x^2 - 2x + 3 = 1

2. Negative Expression: Solve the equation inside the absolute value for when it is negative, negate the expression, and then set it equal to 1:

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

Now, let's solve each equation separately:

1. Positive Expression: x^2 - 2x + 3 = 1

Subtract 1 from both sides:

x^2 - 2x + 2 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-(-2) ± √((-2)^2 - 4 * 1 * 2)) / (2 * 1) x = (2 ± √(4 - 8)) / 2 x = (2 ± √(-4)) / 2

Since the discriminant (4 - 8 = -4) is negative, the solutions will involve complex numbers:

x = (2 + √4i) / 2 or x = (2 - √4i) / 2 x = 1 + i or x = 1 - i

2. Negative Expression: -(x^2 - 2x + 3) = 1

Distribute the negative sign:

-x^2 + 2x - 3 = 1

Add 1 to both sides:

-x^2 + 2x - 3 + 1 = 0

Simplify:

-x^2 + 2x - 2 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-(2) ± √(2^2 - 4 * (-1) * (-2))) / (2 * (-1)) x = (-2 ± √(4 - 8)) / -2 x = (-2 ± √(-4)) / -2

Since the discriminant (-4) is negative, the solutions will involve complex numbers:

x = (-2 + √4i) / -2 or x = (-2 - √4i) / -2 x = 1 - i or x = 1 + i

So, the complete set of solutions to the equation |-x^2 + 2x - 3| = 1 is:

x = 1 + i, 1 - i, 1 + i, or 1 - i