Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Use a calculator to solve the quadratic equation. (Round your answer to three decimal places.) $$ -0.003 x^{2}+0.025 x-0.98=0 $$

Step-by-Step Solution

Answer

The rounded solutions for x are \(x_1 = 8.968\) and \(x_2 = 36.367\)

Step 1: Identify the coefficients

Recognize and write down the coefficients. Here \(a = -0.003\), \(b = 0.025\) and \(c = -0.98\)

Step 2: Substitute into the quadratic formula

Substitute the values of \(a\), \(b\) and \(c\) into the quadratic formula as follows: \(x = \frac{-0.025 \pm \sqrt{(0.025)^2 - 4*(-0.003)*-0.98}}{2*-0.003}\)

Step 3: Computing the Discriminant

Compute the value inside the square root (the discriminant) first: \(D = (0.025)^2 - 4*(-0.003)*-0.98 = 0.000625 + 0.01176 = 0.012385\)

Step 4: Solve for x

Now, compute the two possible solutions using the discriminant: \( x_1 = \frac{-0.025 - \sqrt{0.012385}}{2*-0.003} \) and \( x_2 = \frac{-0.025 + \sqrt{0.012385}}{2*-0.003} \)

Step 5: Round the answers

In the final step, round the answers to three decimal places, using a calculator.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations in the form \(ax^2 + bx + c = 0\). It provides an easy path to finding the roots or solutions of any quadratic equation. The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a\), \(b\), and \(c\) are the coefficients of the equation, where \(a eq 0\). The ± symbol indicates that there are usually two possible values for \(x\), giving two solutions for the equation. This formula is especially useful for equations where factoring is cumbersome or impossible.To use the quadratic formula successfully:

• Identify and write down the values of \(a\), \(b\), and \(c\).
• Plug these values into the formula.
• Simplify carefully, keeping an eye on the signs and the arithmetic involved.
• Remember to evaluate both possible answers using the ±.

Understanding and practicing the quadratic formula is crucial for solving various types of quadratic equations with ease.

Solving Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true. It usually results in two possible solutions. There are several methods for solving quadratic equations, but here we focus on using the quadratic formula.The process involves several key steps:

• Identify the coefficients: Determine \(a\), \(b\), and \(c\) from the quadratic equation.
• Substitute values into the quadratic formula: Substitute these coefficients into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Calculate the discriminant: First solve for the discriminant \(b^2 - 4ac\), which determines the nature of the roots.
• Compute solutions: Insert the discriminant value into the formula and calculate both possible solutions for \(x\).

Most quadratic equations have two solutions, and using the quadratic formula helps find these solutions precisely. Practicing these steps with different equations will build confidence and proficiency.

Discriminant in Quadratic Equations

The discriminant is a part of the quadratic formula under the square root, given by \(D = b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of the quadratic equation.Here's what the discriminant tells us:

• Positive Discriminant (\(D > 0\)): There are two distinct real solutions.
• Zero Discriminant (\(D = 0\)): There is one real solution, which is repeated. The parabola touches the x-axis at a single point.
• Negative Discriminant (\(D < 0\)): There are no real solutions. Instead, there are two complex solutions, and the parabola does not intersect the x-axis.

In the provided exercise, calculating the discriminant was essential in identifying that the equation has two distinct real solutions. Understanding the discriminant helps in quickly assessing what type of solutions to expect, guiding your approach to solving the equation.