Question

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the negatives of the real solutions of the equation \(a x^{2}-b x+c=0\). Assume that \(b^{2}-4 a c \geq 0\)

Short answer

The real solutions of the equation \(a x^{2}+b x+c=0\) are the negatives of the real solutions of the equation \(a x^{2}-b x+c=0\).

Step-by-step solution

- Identify the Quadratic Formula

For a quadratic equation of the form \[ a x^2 + b x + c = 0, \] the solutions can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

- Apply the Quadratic Formula

The solutions of the equation \[ a x^2 + b x + c = 0 \] are: \[ x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \] and \[ x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}. \]

- Solve the Modified Equation

For the equation \[ a x^2 - b x + c = 0, \] apply the quadratic formula again: \[ x = \frac{-(-b) \pm \sqrt{(-b)^2 - 4ac}}{2a}. \]

- Simplify the Modified Equation Solutions

Solving \[ a x^2 - b x + c = 0 \] gives: \[ x_1' = \frac{b + \sqrt{b^2 - 4ac}}{2a} \] and \[ x_2' = \frac{b - \sqrt{b^2 - 4ac}}{2a}. \]

- Compare the Solutions

Notice that \[ x_1' = - x_1 \] and \[ x_2' = - x_2. \] This shows that the solutions of the equation \[ a x^2 - b x + c = 0 \] are the negatives of the solutions of the equation \[ a x^2 + b x + c = 0. \]

Key concepts

quadratic formula

The quadratic formula is a powerful tool that helps us find the solutions for any quadratic equation of the form \(ax^2 + bx + c = 0\). It’s a standard method used to determine the roots of the equation. The formula is written as follows:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula provides two solutions, represented by the \(\pm\) symbol, which means we need to solve for both the positive and negative square roots. This results in two possible values for \(x\), known as the roots or solutions of the quadratic equation. Understanding and applying the quadratic formula is crucial for solving any quadratic equation effectively.

real solutions

In the context of quadratic equations, real solutions refer to the values of \(x\) that satisfy the given equation and result in real numbers. For a quadratic equation to have real solutions, the expression under the square root in the quadratic formula, known as the discriminant, must be non-negative (i.e., \( b^2 - 4ac \geq 0 \)).
Here's why this is important:

• If \( b^2 - 4ac > 0 \): The equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \): The equation has exactly one real solution, which is often called a repeated or double root.
• If \( b^2 - 4ac < 0 \): The equation has no real solutions (the roots are complex numbers).

In our challenge problem example, we assume \( b^2 - 4ac \geq 0 \), ensuring that real solutions exist, making it possible to compare the solutions of two related quadratic equations.

discriminant

The discriminant is the part of the quadratic formula under the square root and is represented by \( \Delta = b^2 - 4ac \). It provides important information about the nature of the solutions of a quadratic equation. Here's a more detailed look:
\[ \Delta = b^2 - 4ac \]
The discriminant helps determine the number and type of solutions as follows:

• \( \Delta > 0 \): The quadratic equation has two distinct real roots.
• \( \Delta = 0 \): The quadratic equation has exactly one real root (roots are coincident).
• \( \Delta < 0 \): The quadratic equation has no real roots (the solutions are complex or imaginary numbers).

When solving quadratic equations, evaluating the discriminant is a crucial step. It instantly tells us whether we should expect real or complex roots, and how many unique solutions the equation will have. Understanding this concept is key to mastering quadratic equations.

symmetry in solutions

In our challenge problem, we explore a fascinating property of quadratic equations: the symmetry in their solutions. Specifically, we show that the real solutions of \(ax^2 + bx + c = 0\) and \(ax^2 - bx + c = 0\) are negatives of each other. Here's what this means:
By applying the quadratic formula to both equations, we derive:

• For \(ax^2 + bx + c = 0 \): \( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) and \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \).
• For \(ax^2 - bx + c = 0 \): \( x_1' = \frac{b + \sqrt{b^2 - 4ac}}{2a} \) and \( x_2' = \frac{b - \sqrt{b^2 - 4ac}}{2a} \).

These solutions reveal:

• \( x_1' = - x_1 \)
• \( x_2' = - x_2 \)

This symmetry arises from how changing the sign of \(b\) in the quadratic equation affects the solutions. This property can be useful in understanding the behavior of quadratic equations and predicting their roots more effectively.