Problem 339 Solve. \(16 p^{3}=24 p^{2}+9 p... [FREE SOLUTION]

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Chapter 7: Problem 339

Solve. \(16 p^{3}=24 p^{2}+9 p\)

Short Answer

Expert verified

The solutions are \(p = 0\), \(p = \frac{3(1 + \sqrt{2})}{2}\), and \(p = \frac{3(1 - \sqrt{2})}{2}\).

Step by step solution

- Set the Equation to Zero

Move all terms to one side of the equation to set it to zero:\(16 p^{3} - 24 p^{2} - 9 p = 0\)

- Factor Out the Common Factor

Identify and factor out the greatest common factor (GCF) which is \(p\):\(p (16 p^{2} - 24 p - 9) = 0\)

- Solve for the Values of p

Solve each factor set equal to zero:1. \(p = 0\)2. Solve \(16 p^{2} - 24 p - 9 = 0\) using the quadratic formula \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), with \(a = 16\), \(b = -24\), and \(c = -9\):\[ p = \frac{24 \pm \sqrt{576 + 576}}{32} = \frac{24 \pm \sqrt{1152}}{32} = \frac{24 \pm 24 \sqrt{2}}{32} = \frac{3(1 \pm \sqrt{2})}{2} \]

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Key Concepts

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

Factoring

Factoring is one of the most essential techniques in solving polynomial equations. It's like breaking down a big problem into smaller, more manageable parts. In this exercise, we started with the equation:
16p^3 - 24p^2 - 9p = 0
Our first step was to factor out the Greatest Common Factor (GCF), which in this case is 'p'. Factoring out 'p' gives us:
p(16p^2 - 24p - 9) = 0
Now our problem is much simpler because we split it into two parts: p = 0 and 16p^2 - 24p - 9 = 0. Factoring helps us break the problem down, making it easier to solve. Always look for the GCF as your first step in factoring!

Quadratic Formula

Once the equation is factored, sometimes we encounter a quadratic equation that we can't easily factor by standard methods. That's where the quadratic formula comes in handy. For our equation 16p^2 - 24p - 9 = 0, we need to solve for 'p'. We use the quadratic formula:
\( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Here, 'a' is 16, 'b' is -24, and 'c' is -9. Plugging in these values, we get:
\( p = \frac{24 \pm \sqrt{576 + 576}}{32} = \frac{24 \pm \sqrt{1152}}{32} = \frac{24 \pm 24 \sqrt{2}}{32} = \frac{3(1 \pm \sqrt{2})}{2} \)
By using the quadratic formula, we are able to find the roots of the quadratic equation, even when it's not straightforward to factor.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number or variable that divides all the terms in an expression without a remainder. It's a crucial step in simplifying equations. For example, in the equation 16p^3 - 24p^2 - 9p, the GCF is 'p'. Factoring out the GCF, we simplify the expression to:
p(16p^2 - 24p - 9) = 0
This simplification allows us to solve for 'p' more easily. Remember, finding the GCF can make a complex problem much simpler. Always check for the GCF before moving on to more complex factoring or applying the quadratic formula.

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91影视

Solve. \(16 p^{3}=24 p^{2}+9 p\)

Short Answer

Step by step solution

- Set the Equation to Zero

- Factor Out the Common Factor

- Solve for the Values of p

Key Concepts

Factoring

Quadratic Formula

Greatest Common Factor

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