LEARN · EXPLAIN · REVISE

Let α and β be the roots. Find the sum of their squares without calculating the individual roots.

Read the idea, work independently, then explain what changed.

TOPIC 01

Quadratic equations and functions

Squaring the sum introduces 2αβ. Leaving that term in would give 25, which is not the sum of the squares.

PREDICT → MOVE → EXPLAIN

Before moving a slider, predict the opening, vertex and intercepts. Then compare the graph with your prediction.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Your turn

x^2-5x+2=0,\qquad \alpha^2+\beta^2=?

Working and explanation

BUILD THE REASONING

Hint 1

Read the sum and product of the roots from the coefficients.

Hint 2

Expand (α + β)², then subtract the cross term.

Worked solution

Use the relationships between roots and coefficients.
$\alpha+\beta=5,\qquad \alpha\beta=2$
Express the required quantity using that sum and product.
$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=25-4=21$

The sum of the squares is 21.

Checks and common pitfalls: Squaring the sum introduces 2αβ. Leaving that term in would give 25, which is not the sum of the squares.

Think first. Reveal a hint when the class is ready.