Lesson 13.4: Applications and Extensions

Property

Any quadratic equation can be expressed in four equivalent forms:

1. Standard form: $ax^2 + bx + c = 0$
2. Factored form: $a(x - r_1)(x - r_2) = 0$ where $r_1, r_2$ are roots
3. Vertex form: $a(x - h)^2 + k = 0$ where $(h, k)$ is the vertex
4. Sum-product form: $x^2 - sx + p = 0$ where $s$ is sum of roots and $p$ is product of roots

Examples

Property

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$, the difference between the roots is:

Property

If $r_1$ and $r_2$ are the roots of $ax^2 + bx + c = 0$, then $ar_1$ and $ar_2$ are the roots of $x^2 + bx + ac = 0$.

Equivalently: The roots of $x^2 + bx + ac = 0$ are $a$ times the roots of $ax^2 + bx + c = 0$.

Property

When solving quadratic equations where the coefficients contain variables or parameters, first identify the coefficients $a$, $b$, and $c$ in standard form $ax^2 + bx + c = 0$, then apply the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Examples