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“Quadratic Equations: Challenge Problems” is available to Standard subscribers.

Harder problems for the keen: equations that turn into quadratics, the sum and the product of the roots, and the greatest or least value of a quadratic, each with worked solutions and a timed paper.

3Lessons11Problems2Python Labs1Question Set1Mock Test

Read the first lesson freeEquations that become quadratic

What's inside

Equations that become quadratic: Many equations that are not quadratic hide a quadratic inside. Put a new letter for $x^2$ , for $x + \frac{1}{x}$ , for a bracket that repeats, or for $2^x$

The chapter itself stays free

This is an extra course for Quadratic Equations. Its main courses are open to everyone:

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The sum and the product of the roots: If

\alpha

and

\beta

are the roots of

ax^2 + bx + c = 0

, then

\alpha + \beta = -\dfrac{b}{a}

and

\alpha\beta = \dfrac{c}{a}

. With these two facts we can build an equation from its roots, and find expressions such as

\alpha^2 + \beta^2

\dfrac{1}{\alpha} + \dfrac{1}{\beta}

without solving. Roots in a given ratio or with a given difference follow too.

Greatest and least values: Completing the square shows the turning point of

y = ax^2 + bx + c

: it is at

x = -\dfrac{b}{2a}

. If

a > 0

that point gives the least value, and if

a < 0

the greatest. We use it for the largest pen from a fixed fence, the highest point of a thrown ball, and the best price for a shop.