Question

In Exercises 31-36, use a calculator to solve the quadratic equation. (Round your answer to three decimal places.) $$ 5.1 x^{2}-1.7 x-3.2=0 $$

Short answer

The solutions to the quadratic equation \(5.1x^{2} - 1.7x - 3.2 = 0\) are x = [Insert the first solution here with three decimal place rounding], x = [Insert the second solution here with three decimal place rounding].

Step-by-step solution

Identifying parameters

Identify the coefficients in the provided quadratic equation. For \(5.1x^{2} - 1.7x - 3.2 = 0\), the coefficients are \(a = 5.1\), \(b = -1.7\), and \(c = -3.2\).

Calculating the Discriminant

Calculate the discriminant, which is the term under the square root in the quadratic formula. It's calculated as \(b^{2} - 4ac\). Substituting the known values, the discriminant is \((-1.7)^{2} - 4 * 5.1 * -3.2\).

Calculating the Roots

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\) to calculate the roots of the equation. Substituting the known values and solving for \(x\), we'll get two solutions (one for '\(+\)' and another for '-').

Rounding the Roots

Round the roots to three decimal places as per the exercise instruction.

Key concepts

Discriminant

In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant plays a crucial role in determining the nature and number of the roots. It is calculated using the formula \(b^2 - 4ac\). This number helps us understand whether the quadratic equation has two distinct real roots, one real root, or no real roots (but possibly complex roots).

• If the discriminant is positive, the quadratic equation will have two distinct real roots.
• If it is zero, there will be exactly one real root (a double root).
• If negative, the equation will have no real roots, but two complex roots.

For our specific example, \((-1.7)^2 - 4 * 5.1 * -3.2\), you need to compute this expression. By squaring \(-1.7\) and then calculating each part of the expression carefully, you uncover critical insight into the potential solutions of the equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the values of \(x\) that satisfy the quadratic equation, regardless of its coefficients and constant term. It is derived from the process of completing the square and can be applied to any quadratic equation as long as you correctly identify \(a\), \(b\), and \(c\) from the equation.
In our example, for the equation \(5.1x^2 - 1.7x - 3.2 = 0\), you start by substituting \(a = 5.1\), \(b = -1.7\), and \(c = -3.2\) into the formula. This will give you two possible values for \(x\), accounting for the '+' and '-' options that the formula contains. Calculating both these values will provide the roots, or solutions, to the equation.

Calculator Usage

Using a calculator effectively can simplify solving quadratic equations, especially when dealing with decimals or complex discriminant calculations. A good calculator can perform precise calculations quickly, ensuring accuracy with operations like squaring numbers or evaluating square roots.
When applying the quadratic formula, input each step clearly into the calculator. Start by computing the discriminant \(b^2 - 4ac\), then calculate the square root of this value. Finally, use these results to evaluate the entire quadratic formula. Many scientific calculators have functions specifically for handling square roots, parentheses, and other necessary operations, which facilitate following the formula correctly.
Remember to finish by rounding the results appropriately as per the exercise requirements, in this case, to three decimal places, to meet precise mathematical standards.