Quadratics: แบบฝึกหัด

Solve the quadratic equation \(6x^2 - 11x - 10 = 0\).

Find the set of values of \(k\) for which the quadratic equation \(x^2 - (k+1)x + (k+4) = 0\) has two distinct real roots.

Find the set of values of the constant \(k\) for which the equation \((x^2 - 2x)^2 - k(x^2 - 2x) + k + 3 = 0\) has exactly four distinct real roots.

Express the quadratic expression \(x^2 + 10x + 1\) in the form \((x+a)^2 + b\) and hence find the minimum value of the expression.

Express \(3x^2 + 12x + 7\) in the form \(a(x+b)^2 + c\), and hence state the coordinates of the vertex of the graph \(y = 3x^2 + 12x + 7\).

Express the quadratic function \(f(x) = x^2 - 10x + 30\) in the form \((x+a)^2 + b\), and hence determine the minimum value of \(f(x)\).

Find the set of values of \(k\) for which the quadratic equation \(kx^2 + (2k+1)x + (k-1) = 0\) has no real roots.

The line \(y = 2x + k\) intersects the curve \(y = x^2 - kx + 3\) at the points \(A\) and \(B\). Find the possible values of the constant \(k\) for which the length of the line segment \(AB\) is \(2\sqrt{5}\).

Find the set of values of the constant \(k\) for which the line \(y = x + k\) intersects the curve \(y = x^2 + 5x + 7\) at two distinct points.

A straight line \(L\) has the equation \(2x - y = 5\). A curve \(C\) has the equation \(y = x^2 - 4x + 9\).

(a) Find the coordinates of the points of intersection, \(A\) and \(B\), of the line \(L\) and the curve \(C\).

(b) Calculate the exact length of the line segment \(AB\).