Question

При каком значении параметра а уравнение (3х-a)^2+(4x +1)^2=(5x- 1)^2 не имеет решений?

Answer 1

X^2 - 2(a+1)x + (3a+7) <= 0
Такое квадратное неравенство не имеет решений, 
если трехчлен слева не имеет решений.
То есть дискриминант должен быть отрицательным.
D/4 = (a+1)^2 - (3a+7) = a^2+2a+1-3a-7 = a^2-a-6 = (a-3)(a+2) < 0
-2 < a < 3
Если а должно быть целым, то a = -1, 0, 1, 2

Answer 2

Solving the Equation (3x-a)^2 + (4x+1)^2 = (5x-1)^2

To find the values of 'a' for which the equation (3x-a)^2 + (4x+1)^2 = (5x-1)^2 has no solutions, we can analyze the discriminant of the quadratic equation.

The given equation is a quadratic equation in terms of 'x'. The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by the expression b^2 - 4ac. If the discriminant is negative, the quadratic equation has no real solutions.

Finding the Discriminant

For the given equation (3x-a)^2 + (4x+1)^2 = (5x-1)^2, we can expand and simplify to obtain a quadratic equation in standard form.

Expanding the given equation: (3x-a)^2 + (4x+1)^2 = (5x-1)^2 9x^2 - 6ax + a^2 + 16x^2 + 8x + 1 = 25x^2 - 10x + 1 25x^2 - 6ax + a^2 + 8x + 1 = 25x^2 - 10x + 1

Simplifying the equation: -6ax + a^2 + 8x = -10x

Rearranging terms: 25x^2 - 6ax + a^2 + 8x = 25x^2 - 10x

Comparing with the standard form ax^2 + bx + c = 0: a = 25, b = -6a, c = a^2 + 8

Calculating the Discriminant

The discriminant D of the quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.

Substituting the values of a, b, and c: D = (-6a)^2 - 4*25*(a^2 + 8)

Simplifying the expression: D = 36a^2 - 100(a^2 + 8) D = 36a^2 - 100a^2 - 800 D = -64a^2 - 800

Conditions for No Solutions

For the given equation to have no solutions, the discriminant D must be negative. Therefore, the condition for no solutions is D < 0.

Conclusion

The equation (3x-a)^2 + (4x+1)^2 = (5x-1)^2 has no solutions when the discriminant D is negative, which occurs when -64a^2 - 800 < 0.

Therefore, the equation (3x-a)^2 + (4x+1)^2 = (5x-1)^2 has no solutions when -64a^2 - 800 < 0.