Algebra

Substitute the given options into the equations. For x=9 and y=4: x+y = 13 and 7+sqrt(36) = 7+6 = 13 (matches). Also, x^2+y^2 = 81+16 = 97, and 133-xy = 133-36 = 97 (matches).

The discriminant D = b^2 - 4ac determines the nature of roots. If D < 0, the roots are complex (not real).

The quadratic formula for ax^2 + bx + c = 0 is x = (-b +/- sqrt(b^2 - 4ac)) / 2a.

The nature of the roots of a quadratic equation ax^2 + bx + c = 0 is determined by the discriminant, which is defined as b^2 - 4ac.

A quadratic equation is of the form ax^2 + bx + c = 0 where a is not 0. Option B is 6x^2 + 7x - 19 = 0, which fits this form.

A quadratic equation has no real roots when the discriminant (b^2 - 4ac) is less than zero.

The equation x^2 + 1/x^2 = 2 can be rewritten as x^4 - 2x^2 + 1 = 0, which is (x^2 - 1)^2 = 0. This is a biquadratic equation, not a quadratic equation in x.

For ax^2 + bx + c = 0, discriminant D = b^2 - 4ac. Here a=1, b=-4, c=a. D = (-4)^2 - 4(1)(a) = 16 - 4a.

The factorization method involves finding two binomials (px+q) and (rx+s) such that their product equals the quadratic expression, allowing the equation to be solved by setting each factor to zero.

The roots are x = (2m +/- sqrt(4m^2 - 4(m^2-1))) / 2 = m +/- 1. So roots are m-1 and m+1. For both to be in (-2, 4), we need -2 < m-1 and m+1 < 4. This gives m > -1 and m < 3.

For roots equal in magnitude and opposite in sign, their sum must be zero. Simplifying the equation and setting the coefficient of x to zero gives the required relation a^2 = 2λ^2. Therefore, option D is correct.

Given alpha + beta = 3 and alpha - beta = 4. Adding: 2*alpha = 7 => alpha = 3.5. Subtracting: 2*beta = -1 => beta = -0.5. The quadratic is (x - 3.5)(x + 0.5) = 0 => x^2 - 3x + 0.5x - 1.75 = 0 => x^2 - 3x - 1.75 = 0. Multiplying by 4: 4x^2 - 12x - 7 = 0.

Factoring the equation x^2 - 2x = 0 gives x(x - 2) = 0. The solutions are x = 0 and x = 2.