Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 6 x=4-x^{2} $$

Step-by-Step Solution

Answer

The solutions of the given quadratic equation are \( x = -3 + \sqrt{8} \) and \( x = -3 - \sqrt{8} \)

Step 1: Convert to Standard Form

Rearrange the terms to convert the equation into standard form: \( -x^2 + 6x - 4 = 0 \)

Step 2: Identify a, b, c

Identify values of a, b and c: a = -1, b = 6, c = -4.

Step 3: Calculate Discriminant

Calculate the discriminant \( D = b^2 - 4ac \) with given values - so it will be \( D = (6)^2 - 4*(-1)*(-4) = 36 - 4 = 32 \).

Step 4: Apply Quadratic Formula

Apply these values into the Quadratic Formula: \( x = \frac{-b±\sqrt{b^2 - 4ac}}{2a} \) which simplifies into \( x = \frac{-6±\sqrt{32}}{2*-1} \)

Step 5: Simplify Further

Simplify the results: it gives two solutions \( x = -3±\sqrt{8} \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomials that have the highest degree of two and are typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable to be solved for. There are several methods to solve such equations like factoring, completing the square, graphing, or using the Quadratic Formula. The most reliable and general method is the Quadratic Formula, which provides a solution for any quadratic equation, even when other methods fail or are difficult to apply.

When given a problem such as solving the equation 6x = 4 - x^2, the first step is to rearrange the terms into standard form. For our example, this would result in -x^2 + 6x - 4 = 0. As you can see, the standard form makes it easier to identify the coefficients of the equation, which are essential for solving it using the Quadratic Formula.

Discriminant of a Quadratic Equation

The discriminant is a key player in the narrative of quadratic equations. It is represented by the symbol 'D' and defined by the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the standard quadratic equation ax^2 + bx + c = 0. The discriminant provides crucial information about the nature of the solutions to a quadratic equation without actually solving it.

• If D > 0, there are two distinct real roots.
• If D = 0, there is exactly one real root (also called a repeated or double root).
• If D < 0, there are no real roots, but rather two complex roots.

In our example, by substituting a = -1, b = 6, and c = -4, the discriminant calculates to be 32, which means the equation has two distinct real roots.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a simple yet powerful tool that is essential for applying various solution methods, including the Quadratic Formula. It is expressed as ax^2 + bx + c = 0. Here, a represents the coefficient of the squared term, b represents the coefficient of the linear term, and c is the constant term. All quadratic equations can be written in this form, and it is crucial for both identifying the coefficients needed to find the discriminant and for substituting the values into the Quadratic Formula.

When looking at an equation like 6x = 4 - x^2, converting it to standard form (-x^2 + 6x - 4 = 0) helps compartmentalize the components of the equation, setting the stage for a systematic approach to find the roots of the quadratic equation. Identifying a as -1, b as 6, and c as -4 is a straightforward process once we have the standard form, facilitating the next steps of solving the equation effectively.