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Quadratic Inequalities (Cambridge O Level Additional Maths)
Revision Note
Quadratic Inequalities
What are quadratic inequalities?
- They are similar to quadratic equations with the "=" replaced by one of <, >, ≤ or ≥
- Just like equations such inequalities should be in a form such that 0 is on one side of the inequality
- e.g.
- Just like equations such inequalities should be in a form such that 0 is on one side of the inequality
- Sketching a quadratic graph is essential to finding the correct solution(s)
- Some modern calculators may be able to solve quadratic inequalities directly
- You could use this to check your answer
- Some modern calculators may be able to solve quadratic inequalities directly
How do I solve quadratic inequalities?
- STEP 1
Rearrange the inequality into quadratic form with a positive squared term
ax2 + bx + c > 0 (>, <, ≤ or ≥) - STEP 2
Find the roots of the quadratic equation
Solve ax2 + bx + c = 0 to get x1 and x2 where x1 ≤ x2 - STEP 3
Sketch the graph of the quadratic and label the roots
As Step 1 makes the -squared term positive it will be -shaped - STEP 4
Identify the region that satisfies the inequality
For ax2 + bx + c > 0 you want the region above the x-axis - the solution will be x < x1 or x > x2
For ax2 + bx + c < 0 you want the region below the x-axis - the solution will be x1 < x < x2
- Be careful:
- avoid multiplying or dividing by a negative number
if unavoidable, “flip” the inequality sign so < → >, ≥ → ≤, etc
- do rearrange to make the x2 term positive
- avoid multiplying or dividing by a negative number
Quadratic inequalities and the discriminant
- The discriminant of the quadratic function is
- It's value indicates the number of (real) roots the quadratic function has
- if there are two roots
- if there is one root (repeated)
- if there are no roots
- The firsts and last of these are quadratic inequalities
- Some questions will require you to use the discriminant to set up and solve a quadratic inequality
- For example: Find the values of such that the equation has no real roots
- Using the discriminant, and for no real roots,
- Using the approach above, this leads to the quadratic inequality in ,
- And using the method above, including sketching a graph, leads to the solutions and
- For example: Find the values of such that the equation has no real roots
Examiner Tip
- Some calculators will solve quadratic inequalities directly and just give you the answer
- Beware!
- make sure you have typed the inequality in correctly
- the calculator may not display the answer in a conventional way
- e.g may be shown as
Both are mathematically correct but the first way is how it would normally be written
- e.g may be shown as
- these questions could crop up on the non-calculator exam paper
- Beware!
Worked example
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Inequalities on Graphs
What are inequalities on graphs?
- Inequalities can be represented on graphs by shaded regions and dotted or solid lines
- These inequalities have two variables, x and y
- Several inequalities are used at once
- The solution is an area on a graph (often called a region and labelled R)
- The inequalities can be linear or quadratic
How do I draw inequalities on a graph?
- Sketch each line or curve
- If the inequality is strict (< or >) then use a dotted line
- If the inequality is weak (≤ or ≥) then use a solid line
- Decide which side of the line satisfies the inequality
- If unsure, choose a coordinate on one side and test it in the inequality
- The origin is an easy point to use
- If it satisfies the inequality then that whole side of the line satisfies the inequality
- For example: (0,0) satisfies the inequality y < x2 + 1 so you want the side of the curve that contains the origin
- If unsure, choose a coordinate on one side and test it in the inequality
Examiner Tip
- Recognise this type of inequality by the use of two variables
- You may have to deduce the inequalities from a given graph
- Pay careful attention to which region you are asked to shade
- Sometimes the exam could ask you to shade the region that satisfies the inequalities this means you should shade the region that is wanted.
Worked example
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