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Chapter 1: Problem 43

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

Short Answer

Expert verified
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 by step solution

01

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
02

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
03

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
04

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
05

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
06

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, \(w = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A Box By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made (see the accompanying figure). If the cardboard is 16 in. long and 10 in. wide, find the dimensions of the resulting box if it is to have a total surface area of \(144 \mathrm{in}\). \(^{2}\)

Solve the equation. $$ \frac{m}{m-2}-\frac{27}{7}=\frac{2}{m^{2}-m-2} $$

Perform the indicated operations and simplify. \(\frac{x}{1-x}+\frac{2 x+3}{x^{2}-1}\)

Perform the indicated operations and simplify. \(2+\frac{1}{a+2}-\frac{2 a}{a-2}\)

Solve the equation by using the quadratic formula. $$ 8(2 m+3)^{2}+14(2 m+3)-15=0 $$
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