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Chapter 3: Problem 38

Find the value of \(3 x^{2}+8 x+5\) for each given value of \(x.\) $$ -\frac{5}{3} $$

Short Answer

Expert verified
The value is 0.

Step by step solution

01

Identify the given value of x

The given value of x is \(-\frac{5}{3}\).
02

Substitute the value of x into the quadratic expression

Replace every instance of x in the expression \(3 x^{2} + 8 x + 5\) with \( -\frac{5}{3} \).
03

Calculate the term involving x squared

Calculate \(3 \left( -\frac{5}{3} \right)^{2} = 3 \left( \frac{25}{9} \right) = \frac{75}{9} = \frac{25}{3}\).
04

Calculate the linear term

Calculate \(8 \( -\frac{5}{3} \) = -\frac{40}{3}\).
05

Calculate the constant term

The constant term is simply 5.
06

Combine all the terms

Finally, combine \( \frac{25}{3} \), \( -\frac{40}{3} \), and 5: \(-\frac{15}{3} + 5 = -5 + 5 = 0 \).

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Key Concepts

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

Substitution in quadratic equations
Substituting values into quadratic equations is a foundational algebraic skill. Here, you are given a specific quadratic expression and a value for the variable. To start, write down the original expression: \(3x^2 + 8x + 5\) and the given value for \(x\) which is \( -\frac{5}{3}\).

To substitute the value of \(x\):
  • Replace every \(x\) in the expression with \(-\frac{5}{3}\).
  • This means rewriting the equation as: \(3 \big( -\frac{5}{3} \big)^2 + 8\big(-\frac{5}{3} \big) + 5\).
Make sure to keep track of negative signs and fractional calculations to avoid mistakes.
Understanding substitution helps to evaluate any expression accurately.
Linear term calculation
When dealing with quadratic equations, calculating the linear term is essential. In the expression \(3x^2 + 8x + 5\), the linear term is the term that contains \(x\) to the first power, which is \(8x\).

Here鈥檚 how to calculate it after substituting \(x = -\frac{5}{3}\):
  • Replace \(x\) with \( -\frac{5}{3} \):\( 8 \big( -\frac{5}{3} \)
  • Multiply the coefficient (8) by the substituted value (-\frac{5}{3}), resulting in: \(-\frac{40}{3} \).
This step is straightforward but involves careful arithmetic to ensure accuracy.
Combining like terms in algebra
Combining like terms is key in simplifying expressions. After substituting and calculating the individual terms, you have:
  • \(\frac{25}{3}\) from the quadratic term
  • \(-\frac{40}{3}\) from the linear term
  • The constant term: 5
To combine these terms:
  • First, convert all terms to a common denominator, if necessary. Here, terms \(\frac{25}{3}\) and \(-\frac{40}{3}\) already have a common denominator.
  • Add and subtract the fractions: \(\frac{25}{3} - \frac{40}{3} + 5 = \frac{-15}{3} + 5 = -5 + 5 = 0 \).
This final step helps simplify the expression to a single value, showing how different components of the expression interact.

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Most popular questions from this chapter

The cost \(y\) of producing \(x\) items is, in some cases, expressed as \(y=m x+b .\) The number \(b\) gives the fixed cost (the cost that is the same no matter how many items are produced), and the number \(m\) is the variable cost (the cost of producing an additional item). It costs \(\$ 2000\) to purchase a copier, and each copy costs \(\$ 0.02\) to make. (a) What is the fixed cost? (b) What is the variable cost? (c) Write the cost equation. (d) What will be the cost of producing \(10,000\) copies, based on the cost equation? (e) How many copies will be produced if the total cost is \(\$ 2600 ?\)

Plot and label each point in a rectangular coordinate system. $$ (5,-4.25) $$

Find each quotient. $$ \frac{-3-5}{2-7} $$

Graph each linear equation. \(x-y=5\)

For each function \(f,\) find \((a) f(2),(b) f(0),\) and \((c) f(-3) .\) See Example 5 $$ f(x)=-3 x+5 $$
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