Sec 3 A-Math Quadratics, Equations & Inequalities topical questions

Express x² + 8x + 3 in the form (x + a)² + b.

State the minimum value of (x − 5)² − 7 and the value of x at which it occurs.

Find the discriminant of 3x² + 2x − 1 and state the number of real roots of 3x² + 2x − 1 = 0.

Solve the inequality x² ≤ 9 and represent your solution on a number line.

Solve the inequality x² − 7x + 10 > 0.

The function f is given by f(x) = x² − 6x + 11.

1. (a)Express f(x) in the form (x + p)² + q.[2]
2. (b)State the minimum value of f(x) and the value of x at which it occurs.[1]
3. (c)Explain why f(x) is always positive.[1]

The line y = 3x + c is a tangent to the curve y = x² + 5x + 4. Find the value of c.

Solve the simultaneous equations y = 2x − 1 and x² + y² = 10.

1. (a)Show that the solutions satisfy 5x² − 4x − 9 = 0.[1]
2. (b)Solve the equation 5x² − 4x − 9 = 0.[2]
3. (c)Write down both solution pairs (x, y).[2]

Find the range of values of k for which the equation kx² + 4x + k = 0 has no real roots.

A ball is thrown so that its height above the ground is h = 15t − 5t² metres, where t is the time in seconds and t ≥ 0.

1. (a)Express h in the form a(t − p)² + q.[2]
2. (b)State the greatest height reached by the ball.[1]
3. (c)State the time at which the greatest height is reached.[1]
4. (d)Find the time when the ball returns to the ground.[1]

The curve y = x² + bx + 4 is a tangent to the line y = 2x, where b is a constant.

1. (a)Show that (b − 2)² = 16.[2]
2. (b)Hence find the two possible values of b.[2]
3. (c)For the positive value of b, state the point of tangency.[1]

Find the set of values of m for which the line y = mx + 4 does not intersect the curve y = x² + 2x + 6. Give exact values.

A seller prices an item at x dollars. The daily demand is (120 − 2x) items, and the daily cost is 500 + 20(120 − 2x) dollars. The daily profit P is revenue minus cost.

1. (a)Show that P = −2x² + 160x − 2900.[2]
2. (b)Express P in the form −2(x − h)² + k.[2]
3. (c)State the price that maximises the daily profit and the maximum profit.[1]

The function y = x² − 4x + c is always positive, where c is a constant.

1. (a)Write down the discriminant of x² − 4x + c in terms of c.[2]
2. (b)Find the range of values of c for which y is always positive.[2]
3. (c)State the smallest integer value of c that works.[1]

The line y = x + k and the curve y = x² − 3x + 5 touch at a single point.

1. (a)Show that the equation x² − 4x + (5 − k) = 0 holds at the contact point.[1]
2. (b)Find the value of k.[2]
3. (c)Find the coordinates of the contact point.[2]