Finding the Roots of a Quadratic Equation

We are asked to find the roots of the given quadratic equation:

$(q - r)x^2 + (r - p)x + (p - q) = 0$

This equation is in the standard form of a quadratic equation, $ax^2 + bx + c = 0$, where:

• $a = q - r$
• $b = r - p$
• $c = p - q$

Analyzing the Coefficients

Let's look at the coefficients $a$, $b$, and $c$. A useful property for quadratic equations is to check the sum of the coefficients:

$a + b + c = (q - r) + (r - p) + (p - q)$

Let's simplify the sum:

$a + b + c = q - r + r - p + p - q$

We can see that the terms cancel out:

$a + b + c = (q - q) + (-r + r) + (-p + p) = 0 + 0 + 0 = 0$

So, the sum of the coefficients is zero.

Using the Property of Roots

For a quadratic equation $ax^2 + bx + c = 0$, if the sum of the coefficients ($a + b + c$) is equal to zero, then one of the roots of the equation is always 1.

Since $a + b + c = 0$ for the given equation, $x = 1$ is one root.

Finding the Second Root

For a quadratic equation $ax^2 + bx + c = 0$ with roots $\alpha$ and $\beta$, we know the following relationships:

• Sum of roots: $\alpha + \beta = -b/a$
• Product of roots: $\alpha \times \beta = c/a$

We already found one root, say $\alpha = 1$. We can use the product of roots formula to find the other root, $\beta$.

$1 \times \beta = \frac{c}{a}$

$\beta = \frac{c}{a}$

Substitute the values of $a$ and $c$ from our equation:

$\beta = \frac{p - q}{q - r}$

So, the second root is $\frac{p - q}{q - r}$.

The Roots

The roots of the equation $(q - r)x^2 + (r - p)x + (p - q) = 0$ are $1$ and $\frac{p - q}{q - r}$.

Comparing these roots with the given options, we find that Option 2 lists the roots as $\frac{p - q}{q - r}$ and $1$.

Revision Table: Quadratic Equation Roots

Additional Information: Solving Quadratic Equations

There are several methods to solve quadratic equations, depending on the specific equation:

• Factoring: If the quadratic expression $ax^2 + bx + c$ can be factored into the form $(px + q)(rx + s)$, the roots are found by setting each factor to zero. This is often the fastest method when applicable.
• Completing the Square: This method involves rewriting the equation in the form $(x - h)^2 = k$ and then taking the square root of both sides. It's a foundational method used to derive the quadratic formula.
• Quadratic Formula: The most general method, the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, always provides the roots for any quadratic equation (as long as $a \neq 0$). The term $b^2 - 4ac$, called the discriminant, tells us about the nature of the roots (real and distinct, real and equal, or complex).
• Checking Sum/Product of Coefficients: As demonstrated in this problem, checking the sum of coefficients ($a+b+c$) and the alternating sum ($a-b+c$) can quickly identify if $1$ or $-1$ are roots, respectively.

Understanding these different methods helps in choosing the most efficient way to solve a given quadratic equation.