Quadratics：练习题

Calculate the discriminant of the quadratic equation \( 3x^2 - 5x + 4 = 0 \) and determine the nature of its roots.

Find the coordinates of the points of intersection of the line \( y = x + 1 \) and the curve \( y = 3x^2 - 4x - 1 \).

The quadratic equation \( kx^2 + (k + 3)x + k = 0 \) has no real roots. Find the set of possible values of the constant \( k \).

Find the value of the discriminant for the quadratic equation \( 2x^2 - 4x + 2 = 0 \) and state the nature of its roots.

Solve the equation \( 4x^4 - 13x^2 + 9 = 0 \) for real values of \( x \).

Solve the quadratic inequality \( 2x^2 - 5x - 3 > 0 \).

Find the coordinates of the points of intersection of the line \( y + 2x = 11 \) and the curve \( xy = 12 \).

Find the set of values of \( m \) for which the line \( y = mx - 5 \) is a tangent to the curve \( y = x^2 + 4 \).

The equation of a curve is such that the sum of a quadratic expression and a linear term is zero.

(a) Solve the quadratic equation \( x^2 - 6x + 5 = 0 \).
(b) By using a suitable substitution, or otherwise, find the real roots of the equation \( x - 6\sqrt{x} + 5 = 0 \).

A curve has the equation \( y = 2x^2 + px + (p - 2) \), where \( p \) is a constant.

(a) Find the set of values of \( p \) for which the curve lies entirely above the \( x \)-axis.
(b) In the case where \( p = 6 \), express the quadratic in the form \( a(x + b)^2 + c \) and hence state the coordinates of the vertex of the curve.