Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by completing the square. $$ 3 x-4=-2 x^{2} $$

Step-by-Step Solution

Answer

The solutions for the given equation, after applying the completing the square method, are: \(x = -\frac{3}{4} \pm \sqrt{\frac{41}{16}}\).

Step 1: Rewrite the equation in standard form

To rewrite the equation in standard form, we need to rearrange the terms so that the quadratic term comes first, then the linear term and the constant term at the end. This will give us: \[ 2x^2 + 3x - 4 = 0 \]

Step 2: Make the leading coefficient 1

In order to make the leading coefficient 1, we will divide both sides of the equation by the leading coefficient, i.e., 2: \[ \frac{1}{2}(2x^2 + 3x - 4) = \frac{1}{2}(0) \] Which yields: \[ x^2 + \frac{3}{2}x - 2 = 0 \]

Step 3: Move the constant term to the right side of the equation

Re-write the equation with the constant term on the right side: \[ x^2 + \frac{3}{2}x = 2 \]

Step 4: Complete the square

To complete the square, we need to add the square of half of the coefficient of the linear term (x), i.e., \(\frac{3}{4}\), to both sides of the equation: \[ x^2 + \frac{3}{2}x + \left(\frac{3}{4}\right)^2 = 2 + \left(\frac{3}{4}\right)^2 \] This simplifies to: \[ x^2 + \frac{3}{2}x + \frac{9}{16} = \frac{41}{16} \]

Step 5: Rewrite the left side as a square

Notice that the left-hand side is now a perfect square. We can rewrite it as: \[ \left(x + \frac{3}{4}\right)^2 = \frac{41}{16} \]

Step 6: Solve for x

Now, we can find the values for x by taking the square root of both sides. Don't forget to include the positive and negative roots: \[ x + \frac{3}{4} = \pm \sqrt{\frac{41}{16}} \] Next, isolate x by subtracting \(\frac{3}{4}\) from both sides: \[ x = -\frac{3}{4} \pm \sqrt{\frac{41}{16}} \] These are the final solutions for x.

Key Concepts

Quadratic Equations

At the heart of many algebra problems lie quadratic equations. These are polynomial equations of the second degree, which means they include an x² term, and take on the standard conceptual format ax² + bx + c where a, b, and c represent constants, with a not equal to zero. The solutions to these equations, often referred to as the 'roots', are the values of x which satisfy the equation ax² + bx + c = 0.

Quadratic equations can be solved using various methods, like factoring, applying the quadratic formula, graphing, or completing the square, as shown in the exercise. Understanding these equations is crucial because they pop up across various areas of mathematics, from geometry to calculus, and even in physics and engineering problems involving projectile motion or optimizing dimensions.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential concept and is expressed as ax² + bx + c = 0. To effectively work with quadratic equations, it's often necessary to manipulate them into this form. It provides a clear structure that allows us to use specific strategies like the quadratic formula or completing the square to find the roots.

The primary strategy employed when using the standard form is gathering all terms on one side of the equal sign, with the x² term leading. The equation 3x - 4 = -2x² in the exercise was rearranged into 2x² + 3x - 4 = 0, reflecting this standard form. Recognizing the proper format can streamline the problem-solving process and lead to a quicker and more accurate identification of solutions.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic expression that can be factored into the square of a binomial. It's the product you get when squaring a binomial of the form (x + d), which equals x² + 2dx + d², where d is half the coefficient of x. Essentially, it’s a trinomial that can be written as (ax + b)², where a and b are real numbers.

In the context of completing the square, as we did in the exercise, finding a perfect square trinomial involves adding a specific value to both sides of the equation that turns the left side of the equation into a perfect square. For the exercise x² + (3/2)x, we added (3/4)² to make it a perfect square trinomial, which could then be simplified to (x + 3/4)². This technique is powerful because it transforms the quadratic equation into a form that can easily be solved by taking square roots, bringing us closer to the solutions for x.