Problem 45 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. All numbers are accurate to at least two significant digits. If \(b^{2}-4 a c>0,\) how do the solutions of \(a x^{2}+b x+c=0\) and \(a x^{2}-b x+c=0\) differ?

Step-by-Step Solution

Answer

The roots differ in position but both equations have two real and distinct roots.

Step 1: Understanding the Discriminant

For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is given by \(b^2 - 4ac\). This discriminant determines the nature of the roots of the equation.

Step 2: Given Condition for First Equation

You are given \(b^{2}-4ac > 0\) for the quadratic equation \(ax^2 + bx + c = 0\). This indicates that the equation has two real and distinct roots.

Step 3: Identifying the Discriminant of Second Equation

Now consider the equation \(ax^2 - bx + c = 0\). The discriminant for this equation can be calculated as \((-b)^2 - 4ac = b^2 - 4ac\), which is the same as the original discriminant for the first equation.

Step 4: Nature of Roots for Both Equations

Since both equations, \(ax^2 + bx + c = 0\) and \(ax^2 - bx + c = 0\), share the same discriminant \(b^2 - 4ac\), and it is greater than zero, they both have two real and distinct roots.

Step 5: Effect of Changing the Sign of 'b'

The change from \(b\) to \(-b\) reflects a horizontal symmetry in the position of the roots around the vertical axis (y-axis). This changes the individual roots' expressions, but since both equations have distinct roots, the exact nature or position of those roots differ due to the opposite signs of \(b\) and \(-b\).

Key Concepts

Discriminant

The discriminant is a crucial part of understanding quadratic equations. It is found in the expression \(ax^2 + bx + c = 0\), and calculated using the formula \(b^2 - 4ac\). The discriminant gives us valuable information about the roots of a quadratic equation, such as whether they are real or complex, and if they are distinct or repeated.

• If the discriminant is greater than zero \((b^2 - 4ac > 0)\), the quadratic equation has two distinct real roots.
• If it equals zero \((b^2 - 4ac = 0)\), the equation has exactly one real root, or in other words, two identical real roots.
• If the discriminant is less than zero \((b^2 - 4ac < 0)\), there are no real roots, but two complex conjugate roots instead.

Knowing the discriminant helps us quickly identify the nature of the roots before even solving the quadratic equation.

Real and Distinct Roots

Real and distinct roots occur when the discriminant \(b^2 - 4ac\) is positive. This indicates that the quadratic equation will intersect the x-axis at two different points.
These points correspond to the solutions of the equation, and each represents a distinct solution for \(x\) in \(ax^2 + bx + c = 0\).
When visualized on a graph, a quadratic equation with real and distinct roots will show a parabola cutting through the x-axis at these two points.
For example, let's consider two equations \(ax^2 + bx + c = 0\) and \(ax^2 - bx + c = 0\) both having a discriminant \(b^2 - 4ac > 0\). They both have real and distinct roots, but the roots are positioned differently on the graph. This is crucial in understanding how changing coefficients affect the graph's and the roots' positions.

Nature of Roots

The nature of roots in a quadratic equation is fundamentally determined by the discriminant.
By analyzing \(b^2 - 4ac\), you can classify the types or nature of the roots without solving the equation.

• When the discriminant is positive \((b^2 - 4ac > 0)\), it confirms that you'll have two distinct real roots.
• A zero discriminant \((b^2 - 4ac = 0)\) implies one real but repeated root, implying the vertex of the parabola touches the x-axis.
• A negative discriminant \((b^2 - 4ac < 0)\) indicates the roots are complex and not real, meaning the parabola doesn't intersect the x-axis.

The concept of the nature of roots reveals the types of solutions to expect from the equation, based on the discriminant's value.

Symmetry in Roots

Symmetry in roots refers to how the roots of quadratic equations may be positioned symmetrically around the y-axis. When considering the equations \(ax^2 + bx + c = 0\) and \(ax^2 - bx + c = 0\), we observe an interesting phenomenon of symmetry.

Flip the sign of \(b\) in these equations. This will essentially mirror the parabola across the y-axis. Even though the discriminant \(b^2 - 4ac\) remains unchanged, the roots of these equations will be reflections of each other across the y-axis.

• This mirror effect stems from the change in the linear coefficient, \(b\), to its negative counterpart, \(-b\), thus altering root positions while maintaining their real and distinct nature.

In essence, this horizontal symmetry reflects a deeper geometric property of quadratic functions, which is highly useful in understanding how solutions are affected by changes to equation coefficients.