The roots α and β

In General Maths you met the sum and product of the roots (see quadratic equations↺). If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then

Further Maths questions use these two facts to find other things about the roots, often when the roots themselves are awkward surds. This lesson shows where the facts come from, then uses them.

Where the two facts come from

If $\alpha$ and $\beta$ are the roots, the quadratic has the brackets $(x - \alpha)$ and $(x - \beta)$. So

• Multiply out the brackets: ${(x - \alpha)(x - \beta) = x^2 - \alpha x - \beta x + \alpha\beta}$.
• Collect the $x$ terms: ${x^2 - (\alpha + \beta)x + \alpha\beta}$.
• Multiply by $a$: ${ax^2 - a(\alpha + \beta)x + a\alpha\beta}$.
• Match the $x$ terms with $bx$: ${b = -a(\alpha + \beta)}$, so ${\alpha + \beta = -\frac ba}$.
• Match the constants with $c$: ${c = a\alpha\beta}$, so ${\alpha\beta = \frac ca}$.

Expressions in α and β

Many expressions stay the same when you swap $\alpha$ and $\beta$, such as $\alpha^2 + \beta^2$. Every one of them can be written using only $\alpha + \beta$ and $\alpha\beta$. Then you put in the sum and the product, and never need the roots.

The key one comes from squaring the sum. The square of side $\alpha + \beta$ is made of $\alpha^2$, $\beta^2$ and two rectangles $\alpha\beta$:

The others you will need:

Each comes from multiplying out. For example, for the cube:

• Multiply out: ${(\alpha + \beta)^3 = \alpha^3 + 3\alpha^2\beta + 3\alpha\beta^2 + \beta^3}$.
• Take out $3\alpha\beta$ from the middle two terms: ${= \alpha^3 + \beta^3 + 3\alpha\beta(\alpha + \beta)}$.
• Take $3\alpha\beta(\alpha + \beta)$ from both sides: ${\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)}$.

Pick an equation and an expression, and compare the answer with the actual roots:

For example, for $2x^2 - 4x - 3 = 0$:

• The sum: ${\alpha + \beta = -\frac{-4}{2} = 2}$.
• The product: ${\alpha\beta = \frac{-3}{2}}$.
• So ${\alpha^2 + \beta^2 = 2^2 - 2\left(-\frac32\right)}$.
• ${= 4 + 3 = 7}$.

For $\alpha - \beta$, find its square first, then take the square root. The answer has $\pm$, because nothing says which root is $\alpha$.

An equation with new roots

Any quadratic with roots $p$ and $q$ can be written

so all you need for the new equation is the sum and the product of the new roots. Work each one out in terms of $\alpha + \beta$ and $\alpha\beta$. At the end, multiply through to clear any fractions.

Your turn