Algebra and functions：練習問題

A quadratic function is given by \( f(x) = x^2 - 8x + 12 \). What are the coordinates of the vertex (turning point) of the graph of \( y = f(x) \)?

Find the coordinates of the two points where the curve \(y = x^2\) intersects the line \(y = 2x + 8\).

Rationalise the denominator of the expression \(\frac{6}{3 - \sqrt{3}}\).

Given that \((x - 2)\) is a factor of the polynomial \(P(x) = 2x^3 + kx^2 - 13x + 6\), find the value of the constant \(k\).

Which of the following describes the roots of the equation \(x^2 + 6x + 9 = 0\)?

Rationalise the denominator of the expression \(\frac{10}{4 - \sqrt{6}}\), giving your answer in the form \(a + b\sqrt{6}\) where \(a\) and \(b\) are rational numbers.

Find the coordinates of the points of intersection of the line \( y = x - 3 \) and the curve \( y = 2x^2 - 5x + 1 \).

Find the value of the constant \(k\) for which the quadratic equation \(3x^2 - 6x + k = 0\) has exactly one real root (a repeated root).

A polynomial function is defined as \( P(x) = 2x^3 - 5x^2 - x + 6 \).

(a) Show that \( (x - 2) \) is a factor of \( P(x) \).

(b) Hence, factorise \( P(x) \) completely into three linear factors.

The curve \( C \) has the equation \( y = x^2 - 4x + 1 \) and the line \( L \) has the equation \( y = 2x + k \), where \( k \) is a constant.

(a) In the case where \( k = -4 \), find the coordinates of the points of intersection between \( L \) and \( C \).

(b) Find the value of \( k \) for which the line \( L \) is a tangent to the curve \( C \).