Problem 107 - 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. \(3 x^{2}-4 x-7=0\)

Step-by-Step Solution

Answer

The solutions are real: \(x = \frac{7}{3}\) and \(x = -1\).

Step 1: Identify the coefficients

For the quadratic equation in standard form, \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 3\), \(b = -4\), \(c = -7\).

Step 2: Calculate the discriminant

The discriminant \(D\) is given by the formula \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(D = (-4)^2 - 4 \times 3 \times (-7) = 16 + 84 = 100\).

Step 3: Analyze the discriminant

Since the discriminant \(D = 100\) is positive, the quadratic equation has two distinct real solutions.

Step 4: Use the quadratic formula

The solutions to the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula, \(x = \frac{-b \pm \sqrt{D}}{2a}\).

Step 5: Substitute into the quadratic formula

Using the values \(b = -4\), \(D = 100\), and \(a = 3\), calculate the solutions: \(x = \frac{-(-4) \pm \sqrt{100}}{2 \times 3} = \frac{4 \pm 10}{6}\).

Step 6: Calculate the solutions

Now solve for the two possible values of \(x\): 1. \(x_1 = \frac{4 + 10}{6} = \frac{14}{6} = \frac{7}{3}\).2. \(x_2 = \frac{4 - 10}{6} = \frac{-6}{6} = -1\).

Step 7: Final conclusion

The solutions to the equation \(3x^2 - 4x - 7 = 0\) are real numbers: \(x = \frac{7}{3}\) and \(x = -1\).

Key Concepts

Discriminant

The discriminant is a vital part of quadratic equations. It helps determine the nature of the roots without actually solving the equation. Let's break this down further to understand its significance:

The discriminant is denoted by the symbol \(D\) and calculated using the formula \(D = b^2 - 4ac\). This formula uses the coefficients \(a\), \(b\), and \(c\) from the standard quadratic equation format \(ax^2 + bx + c = 0\).

Here's how the discriminant works:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), there's exactly one real root, technically termed as a repeated or double root.
• If \(D < 0\), the roots are complex or imaginary; they aren't real numbers.

In our solved example, substituting \(b = -4\), \(a = 3\), and \(c = -7\) into the discriminant formula gives us \(D = 100\), which is positive. Thus, we confirmed the existence of two real roots for this quadratic equation.

Quadratic Formula

The quadratic formula is like a handy tool for precisely finding the roots of any quadratic equation. It's expressed as:\[ x = \frac{-b \pm \sqrt{D}}{2a} \]This formula becomes your go-to solution method after checking the discriminant, \(D\). Here's how you can apply the quadratic formula step-by-step:

• First, identify and substitute \(a\), \(b\), and \(c\) from the quadratic equation of the form \(ax^2 + bx + c = 0\).
• Then you plug the values into the quadratic formula, where \(\pm\) indicates that you'll calculate two possible solutions.

Using the quadratic formula requires calculating the square root of the discriminant \(\sqrt{D}\). It allows you to find both solutions at once by solving \(x = \frac{-b + \sqrt{D}}{2a}\) and \(x = \frac{-b - \sqrt{D}}{2a}\). In our worked example, this gives the precise values of \(x_1 = \frac{7}{3}\) and \(x_2 = -1\).

This formula guarantees that you will find all possible solutions to the quadratic equation, whether they are real or complex.

Real and Complex Roots

Understanding whether the roots of a quadratic equation are real or complex is crucial in mathematics and its various applications.

• **Real Roots:** These are solutions that can be plotted on the number line. They are simple to understand as they correspond to clear points where the quadratic equation crosses the x-axis.
• **Complex Roots:** Complex solutions appear when the equation does not cross the x-axis. Instead of representing real points, they involve the imaginary unit \(i\), where \(i = \sqrt{-1}\).

When the discriminant \(D\) is positive, the quadratic equation offers two real and distinct solutions. These are straightforward to interpret and solve. In contrast, a negative \(D\) would suggest the presence of complex roots, involving non-real components.

In our specific quadratic equation \(3x^2 - 4x - 7 = 0\), we found that \(D = 100\), leading us to two real solutions: \(x = \frac{7}{3}\) and \(x = -1\). This shows that the curve of the equation intersects the x-axis at two points, confirming the presence of real roots.