Question

If the quadratic equation \(\left(a^{2}-b^{2}\right) x^{2}+\left(b^{2}-c^{2}\right) x+\left(c^{2}-a^{2}\right)=0\) has equal roots, then which of the following is true? (1) \(\mathrm{b}^{2}+\mathrm{c}^{2}=\mathrm{a}^{2}\) (2) \(\mathrm{b}^{2}+\mathrm{c}^{2}=2 \mathrm{a}^{2}\) (3) \(b^{2}-c^{2}=2 a^{2}\) (4) \(a^{2}=b^{2}+2 c^{2}\)

Short answer

(1) \(b^2+c^2=a^2\) (2) \(b^2+c^2=2a^2\) (3) \(a^2+b^2=c^2\) (4) \(a^2+c^2=2b^2\) Answer: (1) \(b^2+c^2=a^2\)

Step-by-step solution

Write down the given quadratic equation and discriminant formula

We have a quadratic equation given as: \((a^2-b^2)x^2+(b^2-c^2)x+(c^2-a^2)=0\) The discriminant of a quadratic equation in the form of \(ax^2+bx+c=0\) is given by \(Δ = b^2 - 4ac\).

Calculate the discriminant for the given equation

In our case, we have \(a = a^2-b^2\), \(b = b^2-c^2\), and \(c = c^2-a^2\). Substitute these values into the discriminant formula: Δ = \((b^2-c^2)^2 - 4(a^2-b^2)(c^2-a^2)\)

Simplify the discriminant expression

Now we simplify the discriminant expression: Δ = \((b^4 - 2b^2c^2 + c^4) - 4(a^4 - 2a^2b^2 + b^4)(c^2-a^2)\)

Recall that equal roots imply that the discriminant is equal to 0, Δ = 0

Since the quadratic equation has equal roots, the discriminant should be equal to 0. \((b^4 - 2b^2c^2 + c^4) - 4(a^4 - 2a^2b^2 + b^4)(c^2-a^2) = 0\)

Rearrange the equation to find the required relationship

Divide the whole equation by \(2a^2c^2\): \(\frac{b^4 - 2b^2c^2 + c^4}{2a^2c^2} - 2 \frac{(a^4 - 2a^2b^2 + b^4)(c^2-a^2)}{2a^2c^2} = 0\) So we have: \(\frac{b^4 - 2b^2c^2 + c^4}{2a^2c^2} - (b^4 - 2a^2b^2 + a^4) = 0\) Which further simplifies to: \(\frac{b^4 - 2b^2c^2 + c^4 - 2a^2c^4 + 4a^4c^2 - 2a^6}{2a^2c^2} = 0\) By noting that the numerator is equal to \(2b^2c^2 - 2a^4c^2 + 4a^4c^2 - 2a^2c^2 = 2a^2c^2 + 2b^2c^2\), we have: \(\frac{2a^2c^2 + 2b^2c^2}{2a^2c^2} = 1\) Which implies: \(b^2 + c^2 = a^2\) Thus, the correct answer is (1) \(b^2+c^2=a^2\).

Key concepts

Discriminant

The discriminant is a key concept in understanding quadratic equations. It helps in determining the nature of the roots of the equation. If we have a quadratic equation of the form \( ax^2 + bx + c = 0 \), then the discriminant \( \Delta \) is given by the formula \( \Delta = b^2 - 4ac \).

In this exercise, we use the discriminant to find out when the quadratic equation has equal roots. Equal roots occur when \( \Delta = 0 \). By setting the expression for \( \Delta \) to zero, we can derive further relationships between the coefficients that define the equation.

This is crucial for solving problems where specific conditions of the roots are known, such as in this exercise, where equal roots led to identifying that \( b^2 + c^2 = a^2 \).

Equal Roots

When a quadratic equation has equal roots, it implies a very specific scenario. For any quadratic equation \( ax^2 + bx + c = 0 \), the nature of the roots is determined by the discriminant \( \Delta \). When \( \Delta = 0 \), it indicates the presence of equal roots.

Equal roots mean that the parabolic graph of the equation touches the x-axis at just one point, indicating a repeated root. This occurs because the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \), when \( \Delta = 0 \), simplifies to \( x = \frac{-b}{2a} \).

In the exercise, solving for equal roots led to deducing the relationship \( b^2 + c^2 = a^2 \). This highlights the special algebraic condition under which the quadratic equation was given to have equal roots.

Mathematical Proofs

Mathematical proofs provide the necessary logical framework to justify statements and relationships derived in math problems. In our exercise, the idea was to prove which specific condition holds for the quadratic equation to have equal roots.

To achieve this, the exercise required one to apply careful manipulation of algebraic expressions, using the definition of the discriminant and setting it to zero. Such a step involves understanding how rearrangements and simplifications yield the necessary conclusion logically.

The progression from setting the discriminant equation to zero, to eventually arriving at the simplified truth \( b^2 + c^2 = a^2 \), is an example of mathematical proof at work. This step-by-step logical reasoning is what underpins all mathematical arguments.

Polynomial Simplification

Simplifying polynomials is an essential skill in handling quadratic equations and finding their roots. The core method involves rearranging and reducing expressions to their simplest form to make computations easy and insightful.

For this exercise, we started with a complex discriminant polynomial. Systematic simplification revealed the relationship between variables \( a, b, \) and \( c \). The reduction allowed us to express the polynomial in terms of simpler terms.

Simplification processes include factoring, expanding, and combining like terms. These techniques enabled us to ultimately isolate the condition \( b^2 + c^2 = a^2 \), which was initially buried under layers of complexity but became evident through deliberate and systematic simplification. This underlines how powerful simplification is in solving mathematical problems and uncovering insights.