Problem 24 - Basic Technical Mathematics with Calculus

Question

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$3 x^{2}-25=20 x$$

Step-by-Step Solution

Answer

The solutions are approximately 6.58 and -1.25.

Step 1: Rewrite the Equation

Start by rewriting the equation in standard quadratic form, which is: \[ ax^2 + bx + c = 0 \]The given equation is \[ 3x^2 - 25 = 20x \]Rearrange this equation as follows:\[ 3x^2 - 20x - 25 = 0 \] This puts it in the standard form.

Step 2: Identify Coefficients

Identify the coefficients from the equation in standard form:\[ a = 3, \quad b = -20, \quad c = -25 \] These will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the coefficients into the formula:\[ x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 3 \cdot (-25)}}{2 \cdot 3} \]

Step 4: Calculate the Discriminant

Calculate the discriminant \[ b^2 - 4ac \] using the values obtained:\[ (-20)^2 - 4 \cdot 3 \cdot (-25) = 400 + 300 = 700 \]Because the discriminant is positive, there will be two real roots.

Step 5: Compute the Roots

Substitute the discriminant back into the quadratic formula:\[ x = \frac{20 \pm \sqrt{700}}{6} \]Calculate the two solutions using a calculator:First root:\[ x_1 = \frac{20 + \sqrt{700}}{6} \approx 6.58 \]Second root:\[ x_2 = \frac{20 - \sqrt{700}}{6} \approx -1.25 \]

Step 6: Round the Solutions

The roots need to be rounded to the nearest hundredth as instructed:\[ x_1 \approx 6.58 \]\[ x_2 \approx -1.25 \]These are the approximate solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. This formula is essential because it provides a way to solve any quadratic equation, as long as it is set to zero. Quadratic equations are equations that can be written in the form

• \[ ax^2 + bx + c = 0 \]

The quadratic formula is expressed as:

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

Here, \(a\), \(b\), and \(c\) are coefficients from the equation. The \(-b\) part reverses the sign of the linear coefficient \(b\). The \(\pm\) symbol indicates that there will generally be two solutions: one by adding and the other by subtracting the square root part.
By following this formula, you can determine the x-values (roots) where the quadratic equation equals zero, which is critical for graphing or solving word problems that involve quadratic functions.

Discriminant Calculation

The discriminant is a mathematical expression that tells you about the nature of the roots of a quadratic equation. It is denoted by:

• \[ b^2 - 4ac \]

The value of the discriminant determines whether the quadratic equation has two distinct real roots, one real root (also known as a double root), or two complex roots. When the discriminant is:

• Positive: There are two distinct real roots.
• Zero: There is exactly one real root, also called a repeated or double root.
• Negative: No real roots exist and the solutions are complex numbers.

In our example, the discriminant was calculated as 700, which is positive. This indicates that we have two distinct real roots.

Real Roots

Real roots are simply x-values (or solutions) of a quadratic equation that are real numbers as opposed to imaginary or complex numbers. When the discriminant of a quadratic equation is positive, as we have seen, it means there are two real roots. Real roots can be graphically represented as the points where the parabola (the graph of a quadratic equation) touches or cuts the x-axis. If those roots are distinct, this means the graph intersects the x-axis at two different points.
For the quadratic equation we solved, the roots were approximately 6.58 and -1.25. These values are where the parabola described by the equation \[3x^2 - 20x - 25 = 0\] touches the x-axis. These solutions prove that the equation has real roots.

Standard Quadratic Form

The standard form of a quadratic equation is a crucial step in solving quadratics. It is represented as:

• \[ ax^2 + bx + c = 0 \]

Here, \(a\), \(b\), and \(c\) are known as the coefficients, and they play a pivotal role in the solution process. The goal when given a non-standard quadratic equation—as seen initially with \[3x^2 - 25 = 20x\]—is to rearrange it into the standard form.

• Start by moving all terms to one side of the equation so that you can isolate the zero on the other side.
• This rearrangement enables the use of the quadratic formula effectively.

By converting equations into this form, solving them becomes systematic using the quadratic formula or other methods such as factoring.