Lesson 4: Sums and Products of Roots of a Quadratic

Property

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$:

Examples

Property

For a quadratic $ax^2 + bx + c = 0$:

• If $a = 1$ (monic): sum = $-b$, product = $c$
• If $a \neq 1$ (non-monic): sum = $-\frac{b}{a}$, product = $\frac{c}{a}$

Examples

Property

The solutions of the quadratic equation $a(x - r_1)(x - r_2) = 0$ are $r_1$ and $r_2$. This is called the factored form of the quadratic equation. If you know the two solutions of a quadratic equation, you can work backwards to reconstruct the equation.

Examples

• A quadratic equation has solutions $x=3$ and $x=-6$. The factors are $(x-3)$ and $(x-(-6))$, or $(x+6)$. The equation is $(x-3)(x+6)=0$, which expands to $x^2+3x-18=0$.

• To find an equation with solutions $x=2$ and $x=\frac{1}{4}$, start with factors $(x-2)$ and $(x-\frac{1}{4})$. For integer coefficients, use $(x-2)$ and $(4x-1)$. The equation is $(x-2)(4x-1)=0$, or $4x^2-9x+2=0$.