FP1.2 - Roots of quadratic equations

If $\alpha$ and $\beta$ are the roots of $ax^2+bx+c=0$ with $a\ne0$, their sum and product can be read directly from the coefficients. There is no need to solve the quadratic first.

The result comes from $$a(x-\alpha)(x-\beta)=a\bigl[x^2-(\alpha+\beta)x+\alpha\beta\bigr].$$ Comparing the coefficients of $x$ and the constant term with $ax^2+bx+c$ gives the two relationships.

For $2x^2-3x+7=0$, $$\alpha+\beta=-\frac{-3}{2}=\frac32,\qquad \alpha\beta=\frac72.$$ These values remain valid whether the roots are real or complex.

A symmetric expression is unchanged when $\alpha$ and $\beta$ are swapped. Such expressions can often be rewritten using only $S=\alpha+\beta$ and $P=\alpha\beta$, so the individual roots never need to be found.

For the roots of $2x^2-3x+7=0$, $S=3/2$ and $P=7/2$. Hence $$\alpha^2+\beta^2=\left(\frac32\right)^2-2\left(\frac72\right)=-\frac{19}{4},$$ and $$\alpha^3+\beta^3=\left(\frac32\right)^3-3\left(\frac72\right)\left(\frac32\right)=-\frac{99}{8}.$$

First replace every paired sum, product or reciprocal by $S$ and $P$; then substitute the coefficient values; finally simplify exact fractions. If a higher power appears, build it from a lower symmetric identity rather than expanding unknown roots separately.

To form a quadratic whose roots are transformed versions $u$ and $v$ of $\alpha$ and $\beta$, calculate their new sum $S'=u+v$ and product $P'=uv$. The required monic equation is then $x^2-S'x+P'=0$.

For the roots of $2x^2-3x+7=0$, $S=3/2$ and $P=7/2$. New roots $1/\alpha$ and $1/\beta$ have $$S'=\frac SP=\frac37,\qquad P'=\frac1P=\frac27.$$ Thus $$x^2-\frac37x+\frac27=0,$$ or, with integer coefficients, $$7x^2-3x+2=0.$$

Write the two new roots explicitly; derive their sum and product before inserting numbers; form $x^2-S'x+P'=0$; then multiply through by the least common denominator. A non-zero multiple represents the same quadratic equation, so simplify to integer coefficients when requested.