Problem 112 - 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. \(-x^{2}+7 x-2=0\)

Step-by-Step Solution

Answer

The solutions are real, and they are \(x_1 = \frac{7 - \sqrt{41}}{2}\), \(x_2 = \frac{7 + \sqrt{41}}{2}\).

Step 1: Identify the Discriminant

The discriminant of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). For the equation \(-x^2 + 7x - 2 = 0\), \(a = -1\), \(b = 7\), and \(c = -2\).

Step 2: Calculate the Discriminant

Substitute the values into the discriminant formula: \(D = 7^2 - 4(-1)(-2) = 49 - 8 = 41\).

Step 3: Determine the Nature of the Roots

Since the discriminant \(D = 41\) is greater than zero, the quadratic equation has two distinct real roots.

Step 4: Use the Quadratic Formula

To find the roots, use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a = -1\), \(b = 7\), and \(D = 41\) into the formula.

Step 5: Calculate the Roots

First calculate the square root of the discriminant: \(\sqrt{41}\). Then calculate the roots using the formula: \[ x = \frac{-7 \pm \sqrt{41}}{-2} \]So, the roots are: \[ x_1 = \frac{-7 + \sqrt{41}}{-2}, \quad x_2 = \frac{-7 - \sqrt{41}}{-2} \].

Step 6: Simplify the Roots

Simplify the expressions for the roots:- \(x_1 = \frac{-7 + \sqrt{41}}{-2}\) becomes \(x_1 = \frac{7 - \sqrt{41}}{2}\)- \(x_2 = \frac{-7 - \sqrt{41}}{-2}\) becomes \(x_2 = \frac{7 + \sqrt{41}}{2}\)

Key Concepts

Discriminant

In the study of quadratic equations, the discriminant plays a pivotal role in determining the nature of the roots. It's a part of the quadratic formula, specifically the expression under the square root: \(D = b^2 - 4ac\). Here is what you need to know:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, also called a double root, because both roots are the same.
• If \(D < 0\), the equation has two complex roots, which are conjugates of each other.

Understanding the discriminant helps you foresee the nature of the solutions without actually solving the quadratic equation. It simplifies your approach to these problems and equips you with quicker insights into what kind of answers to expect.

Real and Complex Roots

Roots are the solutions to the quadratic equation, and they can either be real or complex. This depends primarily on the discriminant.

• Real Roots: When your discriminant \(D > 0\) or \(D = 0\), you'll find real numbers. They can be distinct or equal, depending on whether \(D\) is positive or zero.
• Complex Roots: If \(D < 0\), your roots are complex numbers. They have real and imaginary parts and appear as pairs of conjugates.

Identifying these helps you in graphing a quadratic function. Real roots manifest as the points where the graph intersects the \(x\)-axis, while complex roots indicate that the graph does not touch or cross the \(x\)-axis. Recognizing the type of roots you'll encounter sharpens your understanding of quadratic functions and their graphs.

Quadratic Formula

The quadratic formula is a reliable tool for solving quadratic equations. It's written as \(x = \frac{-b \pm \sqrt{D}}{2a}\). Here’s how you can use it step-by-step:

• Identify the coefficients: \(a\), \(b\), and \(c\) in your quadratic equation \(ax^2 + bx + c = 0\).
• Calculate the discriminant \(D = b^2 - 4ac\).
• Substitute \(a\), \(b\), and the computed \(D\) into the formula.
• Solve for \(x\) using the \(\pm\) operator to find both potential solutions.

This straightforward formula enables you to find the roots of any quadratic equation, whether the solutions are real numbers or complex. The quadratic formula not only provides an exact answer but also leverages the discriminant to point out the nature of the roots before you go further into solving them.