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Solved problems for the chapter:

Solving Quadratic Inequalities

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Quadratic inequality

1. Leading coefficient a > 0 (function facing up)

1.1. D > 0 (the function has two real zeros)

1.2. D = 0 (the function has a repeated real zero)

1.3. D < 0 (function has no real zeros)

2. Leading coefficient a < 0 (downward facing function)

2.1. D > 0 (the function has two real zeros)

2.2. D = 0 (function has a repeated real zero)

2.3. D < 0 (function has no real zeros)

3. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:solving_quadratic_inequalities/4859/

Example 1.

Solve the inequality .

Example 2.

Given the quadratic functions and .

Obtain the points where both functions intersect.
Draw their graphs in the same coordinate system.
At what interval are the function values of the function f greater than the value of the function g?

Solve the inequality .

For what real numbers b will the equation:

have two real and distinct solutions?

Given the function , find the values of m for which the function intersects the abscissa axis at two different points.

The function is given.

Draw the graph of f(x).
Solve the inequality .

Solve the inequality .

10.

Obtain the largest domain of the function .

11.

Given the function :

Sketch the graph of f(x).
At what interval is the function negative?

12.

Solve the system of inequalities , .

13.

Solve the inequality .

14.

Solve the inequality .

15.

Solve the inequality .

16.

Solve the inequality .

17.

Solve the inequality .

18.

Solve the inequality .

19.

Solve the inequality .

20.

Solve the inequality .

21.

Solve the system of inequalities:

22.

For what values of n does the line and the parabola have two different points in common?

23.

Solve the system of inequalities , .

24.

Given the function

For what values of m will be the abscissa axis be the tangent to the function f(x)?
For what values of m will the function be above the abscissa axis?

25.

Given the quadratic function:

Obtain and sketct the graph of the function that has its vertex at the point (- 1, 1).
For what value of m does the function has no zero?

26.

For which real number m will the function

not intercept the abscissa,
passes through the coordinate origin.

27.

Draw the set of points (x, y) that satisfy the conditions .

28.

Solve the inequality .

29.

The lines and the parabola are given. Sketch both curves in the same coordinate system, calculate their intersections, and obtain the interval at which the parabola is above the line.

30.

The functions and are given.

Calculate the point of intersections of the functions.
Sketch the graphs of both functions in the same coordinate system.
Obtain the interval at which the function g is above the function f.

31.

The functions and are given.

Calculate the intersections between the functions f and g.
Sketch the graphs of both functions in the same coordinate system.
Obtain the interval at which the function g is above the function f.
Find the equation of the tangent to the parabola which is parallel to the function g.

32.

The quadratic functions and are given.

Calculate the points at which the two functions intersect.
sketch their graphs in the same coordinate system.
For what value of x is f(x) > g(x)?

33.

At what values of the real number x does the function lie below the line ?

34.

The function is given. Find all n for which the vertex of the function f lies above the line .

35.

Solve the inequality . Write the set of solutions as an interval and illustrate it on a number line.