Question

If \(\left(\mathrm{p}^{2}-\mathrm{q}^{2}\right) \mathrm{x}^{2}+\left(\mathrm{q}^{2}-\mathrm{r}^{2}\right) \mathrm{x}+\mathrm{r}^{2}-\mathrm{p}^{2}=0\) and \(\left(\mathrm{p}^{2}-\mathrm{q}^{2}\right) \mathrm{y}^{2}+\left(\mathrm{r}^{2}-\mathrm{p}^{2}\right) \mathrm{y}+\mathrm{q}^{2}-\mathrm{r}^{2}=0\) have a common root for all real values of \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\), then find the common root. (1) \(-1\) (2) 1 (3) 2 (4) \(-2\)

Short answer

Answer: The common real roots are -1 and 1.

Step-by-step solution

Identify the equations

We are given two quadratic equations: 1. \((p^2-q^2)x^2 + (q^2-r^2)x + (r^2-p^2) = 0\) 2. \((p^2-q^2)y^2 + (r^2-p^2)y + (q^2-r^2) = 0\)

Use the quadratic formula to find x in terms of p, q, and r

Applying the quadratic formula to the first equation, we have: \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\) Where \(A = p^2 - q^2\), \(B = q^2 - r^2\), and \(C = r^2 - p^2\). Plugging in the values for A, B, and C, we get: \(x = \frac{-(q^2 - r^2) \pm \sqrt{(q^2 - r^2)^2 - 4(p^2 - q^2)(r^2 - p^2)}}{2(p^2 - q^2)}\)

Use the quadratic formula to find y in terms of p, q, and r

Applying the quadratic formula to the second equation, we have: \(y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\) Where \(A = p^2 - q^2\), \(B = r^2 - p^2\), and \(C = q^2 - r^2\). Plugging in the values for A, B, and C, we get: \(y = \frac{-(r^2 - p^2) \pm \sqrt{(r^2 - p^2)^2 - 4(p^2 - q^2)(q^2 - r^2)}}{2(p^2 - q^2)}\)

Compare x and y

Now, observe that both x and y have the same denominators, so we can compare their numerators: \(-(q^2 - r^2) \pm \sqrt{(q^2 - r^2)^2 - 4(p^2 - q^2)(r^2 - p^2)} = -(r^2 - p^2) \pm \sqrt{(r^2 - p^2)^2 - 4(p^2 - q^2)(q^2 - r^2)}\)

Evaluate the discriminant

After comparing the numerators, we can equate the discriminants to find the common root: \((q^2 - r^2)^2 - 4(p^2 - q^2)(r^2 - p^2) = (r^2 - p^2)^2 - 4(p^2 - q^2)(q^2 - r^2)\) We observe that the discriminants are the same, which means that x and y must have the same roots.

Test the given options

We can now test the given options to see which one is a common root for both equations. 1. For x = y = -1 and p = q = r = -1, both equations are true. 2. For x = y = 1 and p = q = r = 1, both equations are true. Hence, both options (1) and (2) are common roots, i.e., -1 and 1.

Key concepts

Common Roots

Having a common root between two quadratic equations means that there is at least one solution that satisfies both equations simultaneously. In other words, the value that solves one equation must also solve the other. This is an interesting concept because when two equations share a common root, it often reveals deeper relationships between the coefficients or structures of the equations.

• For example, if equations are structured similarly or have coefficients that relate in specific ways, you'll often find common roots.

To find common roots, one can equate the quadratic solutions derived from the quadratic formula. If the discriminants of both equations equate, it indicates that each equation can be solved alike under the given conditions.
Ultimately, testing the given root options confirms which values satisfy both equations simultaneously. In the exercise above, x = y = -1 and x = y = 1 satisfied the conditions for both quadratic equations, making them common roots.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It enables us to solve for x, the variable, using the coefficients a, b, and c. The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is derived from the method of completing the square and is particularly useful because it provides exact solutions, even when a quadratic cannot be factored easily.

• If we use the quadratic formula, it's essential to pay attention to the signs before b, as well as within the discriminant.
• The plus-minus (\(\pm\)) symbol indicates that there could be two possible solutions for x.

In the given exercise, applying the quadratic formula to both equations helped in evaluating potential common roots. By plugging coefficients into the formula, we found expressions for x and y that could potentially be equal in certain scenarios.

Discriminant

The discriminant in the quadratic formula is the expression under the square root: \(b^2 - 4ac\). It tells us about the nature of the roots of the quadratic equation. Here's what it reveals:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root, meaning the solution touches the x-axis at a single point (also known as repeated or double root).
• If the discriminant is negative, there are no real roots, implying the roots are complex or imaginary.

In our exercise, both the discriminants were equated to investigate common roots. By ensuring \((q^2 - r^2)^2 - 4(p^2 - q^2)(r^2 - p^2)\) and \((r^2 - p^2)^2 - 4(p^2 - q^2)(q^2 - r^2)\) were equal, we ensured the potential for a shared common root. This comparison stage is crucial since any variance might imply a difference in root values or nature, thereby affecting the common root deduction.