Roots of quadratic equations

What you’ll learn

• Connect coefficients to sums and products of roots.
• Form quadratics with transformed roots.

Understand the idea

If $\alpha$ and $\beta$ solve $ax^2+bx+c=0$, expanding $a(x-\alpha)(x-\beta)$ gives their sum and product without solving individually. These symmetric expressions often simplify algebra involving the roots.

To create an equation with new roots, calculate their sum $S$ and product $P$, then write $x^2-Sx+P=0$. For a quadratic with real coefficients and negative discriminant, the two roots form a conjugate pair.

Key formulae

$$\alpha+\beta=-\frac ba,\qquad\alpha\beta=\frac ca$$

$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$

$$x^2-Sx+P=0$$

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Worked example

The roots of $x^2-5x+3=0$ are $\alpha,\beta$. Form an equation with roots $\alpha+2,\beta+2$.

1. The original sum is $5$ and product is $3$.
2. The new sum is $5+4=9$.
3. The new product is $\alpha\beta+2(\alpha+\beta)+4=3+10+4=17$.

Answer: $x^2-9x+17=0$.