Problem 24 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. If there are no real roots, state this as the answer. \(3 x^{2}-25=20 x\)

Step-by-Step Solution

Answer

The solutions are approximately \( x \approx 7.74 \) and \( x \approx -1.08 \).

Step 1: Simplify the Equation

Start by rearranging the equation to set it to zero. Move all terms to one side:\[ 3x^2 - 20x - 25 = 0 \]

Step 2: Identify Quadratic Formula Components

The equation is now in the form \( ax^2 + bx + c = 0 \) where \( a = 3 \), \( b = -20 \), and \( c = -25 \).

Step 3: Apply Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \):\[ x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 3 \cdot (-25)}}{2 \cdot 3} \]

Step 4: Calculate Discriminant

Calculate the discriminant part of the quadratic formula, \( b^2 - 4ac \):\[ (-20)^2 - 4 \, (3) \, (-25) = 400 + 300 = 700 \]

Step 5: Compute the Roots

Since the discriminant is positive, we have real roots. Solve for \( x \):\[ x = \frac{20 \pm \sqrt{700}}{6} \]Calculate the two solutions:1. \( x = \frac{20 + 26.46}{6} \approx 7.74 \)2. \( x = \frac{20 - 26.46}{6} \approx -1.08 \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions, or 'roots', of quadratic equations. A quadratic equation is typically represented in the standard form:\[ ax^2 + bx + c = 0 \]This is where the quadratic formula comes into play, as it allows you to quickly and efficiently find the values of \( x \) by plugging in the coefficients \( a \), \( b \), and \( c \) from your equation. The formula itself is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]With the quadratic formula, you're equipped to deal with any quadratic equation, whether it has real or complex roots. Remember that the terms "\( -b \pm \)" signify that you will need to perform two calculations to find the two possible solutions for \( x \).Using this formula requires accurate calculations, which makes it very useful to know how to use a scientific calculator effectively. Having a strong grasp of this formula and its application is crucial when dealing with quadratic equations.

Real Roots

Real roots refer to the solutions of the quadratic equation where the values obtained from the formula are real numbers. Quadratic equations can have either real roots or complex numbers as roots, depending on the value of the discriminant (which we'll cover in the next section).When solving quadratic equations, you'll often encounter scenarios with either:

• Two distinct real roots
• One real root (this occurs when they are repeated or equal)
• No real roots (meaning the solutions are complex)

In the context of the equation \( 3x^2 - 20x - 25 = 0 \), the discriminant was calculated to be positive, indicating that the solutions to this particular quadratic equation are real. This means that the graph of the corresponding quadratic function intersects the x-axis at two points, which are approximately \( x \approx 7.74 \) and \( x \approx -1.08 \). This is key for visualizing and understanding the solutions in practical terms.

Discriminant

The discriminant is part of the quadratic formula under the square root, specifically the expression \( b^2 - 4ac \). It holds significant importance because it tells us about the nature of the roots without actually computing them entirely. The value of the discriminant determines whether the roots are real or complex:

• If \( b^2 - 4ac > 0 \): There are two distinct real roots.
• If \( b^2 - 4ac = 0 \): There is exactly one real root (also known as a repeated or equal root).
• If \( b^2 - 4ac < 0 \): The roots are complex (not real).

For the equation \( 3x^2 - 20x - 25 = 0 \), we found that the discriminant is 700. This positive value indicates the presence of two real and distinct roots. This understanding can be very helpful when you're tasked to predict the type of solutions without solving the entire equation, saving you time and effort in complex problems.

Calculator Usage

Using a calculator efficiently, especially in solving quadratic equations, can significantly simplify your problem-solving process. Here’s why and how you should streamline calculator usage:First, ensure that you are comfortable with basic and scientific calculator functions. For calculating roots using the quadratic formula, follow these tips:

• Use parentheses wisely to maintain the structure of your expressions.
• Double-check your input to avoid common errors with signs and parentheses.
• Utilize the square root function diligently to solve accurately, especially when handling decimals.

In our example of the equation \( 3x^2 - 20x - 25 = 0 \), careful calculator use allows you to compute the discriminant, \( 700 \), and then further proceed to evaluate \( \sqrt{700} \) accurately. The calculator can aid in evaluating the complete quadratic formula:\[ x = \frac{20 \pm 26.46}{6} \] to yield the solutions precisely, \( 7.74 \) and \( -1.08 \). As an enhanced strategy, use the calculator to verify each step's results to minimize the likelihood of errors, ensuring accurate solutions every time.