Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by using the quadratic formula. $$ 6 w-13 \sqrt{w}+6=0 $$

Step-by-Step Solution

Answer

The given equation is \(6w - 13\sqrt{w} + 6 = 0\). After expanding, simplifying, and factoring, we get the equation \(w(36w + 169 - 156\sqrt{w}) = 0\). We find one solution, \(w = 0\), but the other solution cannot be found using the quadratic formula without further simplification. So the short answer is \(w = 0\).

Step 1: Rewrite equation in quadratic form

Before we get started, let's rewrite the equation so that it is in quadratic form. We're given: \(6w - 13\sqrt{w} + 6 = 0\) Square both sides of the equation to eliminate the square root term: \((6w - 13\sqrt{w} + 6)^2 = 0^2\) Step 2: Expand the equation

Step 2: Expanding the squared equation

Now let's expand the left-hand side of the equation using the binomial expansion formula: \((a-b+c)^2 = a^2 - 2ab + 2ac + b^2 - 2bc + c^2\) So we have: \((6w)^2 - 2(6w)(13\sqrt{w}) + 2(6w)(6) + (13\sqrt{w})^2 - 2(13\sqrt{w})(6) + 6^2 = 0\) Step 3: Simplify the expanded equation

Step 3: Simplify the squared equation

Next, we simplify the equation by calculating the individual terms: \(36w^2 - 156w\sqrt{w} + 72w + 169w - 156w + 36 = 0\) Step 4: Gather like terms

Step 4: Combining like terms

Now we can combine the like terms in the equation: \(36w^2 + 169w - 156w\sqrt{w} = 0\) Step 5: Factor out the GCF

Step 5: Factor out the GCF

Factor out the greatest common factor (GCF) of the terms: \(w(36w + 169 - 156\sqrt{w}) = 0\) Step 6: Solve for w

Step 6: Solving for w

Now we have two possible solutions for the equation: 1) If \(w = 0\), then the equation holds true because: \(0(36(0) + 169 - 156\sqrt{0}) = 0\) 2) If the term inside the parentheses is equal to 0: \(36w + 169 - 156\sqrt{w} = 0\) Since the equation is non-standard quadratic at this point, it can't be solved using the quadratic formula without further simplification. However, we have one solution, \(w = 0\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the standard form \(ax^2 + bx + c = 0\). It is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula provides the solutions (roots) for any quadratic equation, provided that the equation is in the correct form. However, when dealing with non-standard quadratics, adjustments may be necessary before applying the formula. The discriminant, denoted as \(b^2 - 4ac\), plays a crucial role in determining the nature of the roots, indicating whether they are real or complex. In cases where it's non-positive, the roots are often complex.

Solving Non-standard Quadratics

Non-standard quadratics, such as the equation \(6w - 13\sqrt{w} + 6 = 0\), require extra steps as they do not fit the typical quadratic formula format. The key is to manipulate the equation into a recognizable form.

• First, identify any square root or non-linear components.
• Perform algebraic operations, such as squaring, to simplify.
• Advance through the setup phase to transform it into a standard quadratic format if possible.

Often, these equations need additional solving methods beyond the quadratic formula. In this problem's solution, squaring both sides helped eliminate the square root, leading towards a form amenable to further simplification.

Binomial Expansion

Binomial expansion is a technique used to expand expressions raised to a power, significantly aiding in this problem when simplifying terms. The binomial formula expands expressions such as \((a+b)^2\) using:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• For three terms as in \((a-b+c)^2\), use each pair of terms product in the expansion: \(a^2 - 2ab + 2ac + b^2 - 2bc + c^2\).

This approach was applied in the problem by expanding the terms before simplifying them into a recognizable quadratic expression. Correctly executing this expansion is critical for solving complex algebraic problems.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. It is essential for simplifying expressions and equations. In this context, it allowed us to simplify the equation further by factoring common elements.

• Identify common factors in the equation's terms.
• Factor out these elements to simplify the problem.
• This step can sometimes offer clear paths to potential solutions.

In this equation, extracting the GCF from terms resulted in a simplified view, helping to solve for \(w\). Identifying the GCF is a fundamental skill as it can make complex algebraic expressions more approachable.