Question

Find the roots of each quadratic by any of the methods shown in this section. Keep three significant digits. For some, use more than one method and compare results. Challenge Problems. $$6.47 x-338=205-3.73 x^{2}$$

Short answer

The roots of the equation are approximately x = -8.81 and x = 5.31 after applying the quadratic formula.

Step-by-step solution

Rearrange in standard quadratic form

Bring all terms to one side to set the quadratic equation to the standard form of ax^2 + bx + c = 0. Combine like terms:

Identify coefficients

Identify coefficients a, b, and c in the standard form of the quadratic equation:

Use the quadratic formula

Apply the quadratic formula to find the roots of the equation. The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / (2a):

Calculate the discriminant

Calculate the discriminant (∆ = b^2 - 4ac) to determine the nature of roots:

Solve for the roots using the discriminant

Find the two roots of the quadratic equation using the discriminant in the quadratic formula:

Key concepts

Quadratic Formula

The quadratic formula offers a straightforward technique for finding the roots of any quadratic equation. Given a quadratic equation in the form of \(ax^2+bx+c=0\), the formula specifies the roots as \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where a, b, and c are the coefficients of the equation.

These coefficients represent the quadratic coefficient (\(a\)), the linear coefficient (\(b\)), and the constant term (\(c\)), respectively. It is important to remember that \(a\) cannot be zero, or the equation would not be quadratic. The term under the square root, \(b^2-4ac\), is known as the discriminant, which determines the nature of the roots.

Using the quadratic formula is particularly helpful when factoring is not feasible or when the equation cannot be easily simplified. By inputting the coefficients into the formula, students can efficiently solve quadratics and gain insight into the properties of their solutions.

Discriminant

The discriminant is a key component in the study of quadratic equations and is represented by \(\Delta = b^2 - 4ac\). It is part of the quadratic formula and plays a crucial role in determining the number and type of roots that a quadratic equation will have.

If \(\Delta > 0\), the quadratic equation has two distinct real roots. When \(\Delta = 0\), it implies that there are two real and identical roots, often referred to as repeated roots. However, if the discriminant is negative (\(\Delta < 0\)), the equation possesses two complex roots, which are not real numbers.

Understanding the discriminant provides insight into whether the roots can be represented on a real number line or if they involve imaginary numbers. This is particularly useful in various applications, such as physics and engineering, where real and complex roots can have different practical meanings.

Standard Quadratic Form

A quadratic equation is considered to be in its standard quadratic form when it is written as \(ax^2 + bx + c = 0\), where a, b, and c are coefficients and \(a \eq 0\).

Placing an equation into standard form is an essential first step before applying methods such as factoring, completing the square, or using the quadratic formula for finding roots. To transform an equation into this form, terms need to be rearranged and like terms combined, ensuring that the \(x^2\) term is positive.

Moving Terms and Combining Like Terms

In the context of our problem, \(6.47x - 338 = 205 - 3.73x^2\), we move all terms to the left side and combine like terms to obtain \(3.73x^2 + 6.47x - 543 = 0\). This puts the equation into the necessary format to then identify the coefficients: \(a = 3.73, b = 6.47, and c = -543\). Once in standard form, we're equipped to leverage the powerful tools provided by the quadratic formula and discriminant to solve for \(x\).