Problem 109 - Calculus for Biology and Medicine

Question

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(-x^{2}+2 x-1=0\)

Step-by-Step Solution

Answer

Solutions are real and repeated: \( x = 1 \).

Step 1: Identify the coefficients

For the quadratic equation in the form \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = -1 \), \( b = 2 \), and \( c = -1 \).

Step 2: Calculate the discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Substitute the coefficients: \( \Delta = 2^2 - 4(-1)(-1) = 4 - 4 = 0 \).

Step 3: Determine nature of roots

Since the discriminant \( \Delta = 0 \), the quadratic equation has one real, repeated solution (a double root).

Step 4: Apply the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the coefficients and discriminant: \( x = \frac{-2 \pm \sqrt{0}}{2(-1)} = \frac{-2 \pm 0}{-2} \).

Step 5: Calculate the roots

Since \( \sqrt{0} = 0 \), the quadratic equation simplifies to \( x = \frac{-2}{-2} = 1 \). Thus, the root is \( x = 1 \). It is a repeated root.

Key Concepts

Discriminant

When dealing with quadratic equations, the discriminant is a useful concept as it helps us determine the nature of the roots without having to solve the equation fully. The discriminant, denoted by \( \Delta \), is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

In this formula, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). By calculating \( \Delta \), you can predict whether the roots are real or complex:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), it has one real double root (repeated root).
• If \( \Delta < 0 \), the roots are complex (not real).

In our example, the discriminant is calculated as \( \Delta = 4 - 4 = 0 \). This tells us the quadratic equation has one real, repeated solution.

Real Roots

In the context of quadratic equations, real roots are the

• values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \)

when the discriminant is non-negative. When \( \Delta = 0 \), the equation has a double root, which means a single real number that satisfies the equation twice.
In our computation, the discriminant is 0, indicating that there is one real root.

• This root is considered a repeated or double root as the quadratic equation touches the \( x \)-axis but does not cross it.

In essence, the real roots are points where the parabola (graph of the quadratic equation) intersects or merely touches the \( x \)-axis.

Quadratic Formula

The quadratic formula is a vital tool for finding the roots of any quadratic equation. It is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula directly uses the coefficients \( a \), \( b \), and \( c \), along with the discriminant \( \Delta \). Using the quadratic formula, you can find the precise roots of the equation regardless of whether they are real or complex.
In the provided example, substituting the values of \( a = -1 \), \( b = 2 \), and \( c = -1 \) into the formula gives us:

• \( x = \frac{-2 \pm \sqrt{0}}{-2} = \frac{-2}{-2} = 1 \)

As illustrated, the zero in the square root indicates both the presence of a single solution and aligns with our previous finding that \( \Delta = 0 \).

Double Root

A double root in the realm of quadratic equations occurs when the discriminant, \( \Delta \), equals zero. This means the quadratic equation touches the \( x \)-axis at exactly one point.

• The double root signifies that this single root satisfies the equation twice.

From a graphical perspective:

• The curve contacts the \( x \)-axis at one point but does not cross it.

In our specific example, after applying the quadratic formula and simplifying, we found that \( x = 1 \). This means 1 is a repeated or double root of the equation \(-x^2 + 2x - 1 = 0\). Understanding double roots helps in appreciating how the solutions align with the equation's graphical representation. It vividly highlights how small changes can drastically alter the nature of solutions in quadratic equations.