Inequalities path · I04

Before this lesson: Algebraic identities and factorisation, Inequality rules and signs

Your goal: Turn a desired bound into a sum of nonnegative terms.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Sum of Squares Method for Inequalities: the key idea

Every real square is nonnegative. An inequality can therefore be proved by rewriting the difference between its two sides as a sum of squares with nonnegative coefficients. Begin by predicting equality: the squares in a useful decomposition must all vanish there. Completing the square handles one variable; identities such as a²+b²−2ab=(a−b)² handle symmetric expressions. A decomposition is a proof only after expansion verifies that it equals the required difference. Equality requires every positive-weight square to vanish simultaneously.

A worked example

Find the minimum of x²−6x+13 over the reals.

Complete the square: x²−6x+13=(x−3)²+4. The square is nonnegative, so the value is at least 4, attained exactly at x=3.

Your turn: change one thing

Prove a²+b²+c²≥ab+bc+ca for real a,b,c.

Try this on paper before opening the explanation.

Compare your reasoning

Twice the difference is (a−b)²+(b−c)²+(c−a)²≥0. Equality requires a=b=c.

Pause and check

A trap to avoid: Proving a different expression or missing when all squares vanish together.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove a²+b²≥2ab for every pair of real numbers and state equality.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Subtract 2ab: a²+b²−2ab=(a−b)²≥0. Equality holds if and only if the square vanishes, namely a=b. The variables need not be positive for this particular inequality.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. What is the minimum possible value of a real square?

Foundation

  1. -1
  2. 1
  3. 2
  4. 0
Hint

Squares cannot be negative.

Answer and reasoning

0. The value 0 is attained by squaring 0.

2. Complete the square: x²−4x+4 equals what?

Foundation

  1. x²−4
  2. (x−4)²
  3. (x−2)²
  4. (x+2)²
Hint

Expand the proposed square.

Answer and reasoning

(x−2)². (x−2)²=x²−4x+4.

3. Find the minimum of x²−8x+19.

Core

  1. 4
  2. 19
  3. 3
  4. 0
Hint

Complete the square around x=4.

Answer and reasoning

3. x²−8x+19=(x−4)²+3, so the minimum is 3.

4. When does a²+b²=2ab hold for real a,b?

Core

  1. a=−b always
  2. a+b=1
  3. ab=1
  4. a=b
Hint

The difference is (a−b)².

Answer and reasoning

a=b. Equality is equivalent to a−b=0.

5. Which expression equals 2(a²+b²+c²−ab−bc−ca)?

Stretch

  1. (a−b)²+(b−c)²+(c−a)²
  2. (a+b+c)²
  3. (a−b−c)²
  4. a²+b²+c²
Hint

Expand and count each square term twice.

Answer and reasoning

(a−b)²+(b−c)²+(c−a)². The three differences contribute 2a²+2b²+2c²−2ab−2bc−2ca.

6. A proposed SOS proof has a negative coefficient on one square. Is nonnegativity automatic?

Stretch

  1. Only for integers
  2. Only for three variables
  3. No
  4. Yes, all squares force any linear combination positive
Hint

The coefficient changes the sign of the contribution.

Answer and reasoning

No. A negative weighted square can be negative. Another argument is required.

7. For real x, (x−2)² is always what?

Foundation

  1. Negative
  2. An integer
  3. Nonnegative
  4. Strictly positive
Hint

It can equal zero at x=2.

Answer and reasoning

Nonnegative. Every real square is at least zero.

8. Find the minimum of x²+4x+9.

Core

  1. 4
  2. 0
  3. 5
  4. 9
Hint

Complete the square.

Answer and reasoning

5. x²+4x+9=(x+2)²+5, with minimum 5.

9. In a sum of positive-weight squares, equality to zero requires what?

Stretch

  1. Every variable is positive
  2. The weights sum to one
  3. Every square is zero
  4. Only one square is zero
Hint

No positive term can cancel another.

Answer and reasoning

Every square is zero. All summands are nonnegative and none can offset a positive one.

Choose your next step

Continue to Arithmetic, geometric and harmonic means. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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