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Chapter 5: Problem 19

If \(P(x)=x^{2}+x+1\) and \(Q(x)=5 x^{2}-1,\) find the following. See Examples 4 and \(5 .\) $$ Q\left(\frac{1}{4}\right) $$

Short Answer

Expert verified
Q(1/4) = -11/16.

Step by step solution

01

Identify the Function

In this exercise, we are given the function \(Q(x) = 5x^2 - 1\). Our task is to evaluate this function at a specific input value.
02

Substitute the Value

To find \(Q\left(\frac{1}{4}\right)\), substitute \(x = \frac{1}{4}\) into the function. This means we replace every \(x\) in \(Q(x)\) with \(\frac{1}{4}\).
03

Calculate the Square

Calculate \(\left(\frac{1}{4}\right)^2\). The rule for squaring a fraction is to square both the numerator and the denominator. Thus, \(\left(\frac{1}{4}\right)^2 = \frac{1}{16}\).
04

Multiply by 5

Substitute \(\frac{1}{16}\) back into the function and multiply it by 5: \(5 \times \frac{1}{16} = \frac{5}{16}\).
05

Subtract 1

Subtract 1 from the result obtained in the previous step: \(\frac{5}{16} - 1\). Convert 1 to a fraction with a denominator of 16, which is \(\frac{16}{16}\).
06

Perform the Subtraction

Perform the subtraction \(\frac{5}{16} - \frac{16}{16}\). This results in \(-\frac{11}{16}\).
07

Final Result

Thus, the value of \(Q\left(\frac{1}{4}\right)\) is \(-\frac{11}{16}\).

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Key Concepts

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

Substitution Method
The substitution method is a key concept in mathematics, especially useful in evaluating functions and solving equations. It involves replacing a variable with a specific value or expression. This method allows for simplifying complex equations or evaluating functions at specific points.
  • In this exercise, you're given the function \(Q(x) = 5x^2 - 1\) and you need to find \(Q\left(\frac{1}{4}\right)\).
  • Substitute \(x = \frac{1}{4}\) into the equation wherever you see \(x\). This yields: \(5 \left(\frac{1}{4}\right)^2 - 1\).
By substituting, you transform a general expression that depends on \(x\) into a specific value that can be calculated directly.
Squaring Fractions
Squaring fractions involves raising a fraction to the power of two. This is done by individually squaring the numerator and the denominator. For instance, if you have the fraction \(\frac{a}{b}\), then squaring it results in \(\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}\).
When you substitute \(\frac{1}{4}\) into the function, you need to square it:
  • Square the numerator: \(1^2 = 1\).
  • Square the denominator: \(4^2 = 16\).
  • The result is \(\left(\frac{1}{4}\right)^2 = \frac{1}{16}\).
This simple operation is crucial for evaluating the function accurately, as it affects the calculations that follow.
Fractions Subtraction
Subtracting fractions requires a common denominator, allowing you to directly subtract the numerators.
Here, you perform the subtraction \(\frac{5}{16} - 1\). First, convert the integer 1 into a fraction with a common denominator:
  • Change 1 to \(\frac{16}{16}\) so it matches the denominator of \(\frac{5}{16}\).
Now, subtract:
  • Subtract the numerators: \(5 - 16 = -11\).
  • Keep the denominator the same: \(16\).
Thus, \(\frac{5}{16} - \frac{16}{16} = -\frac{11}{16}\). Understanding fractions subtraction is essential for achieving the correct final result in any algebraic operation involving fractions.

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Most popular questions from this chapter

Multiply each expression. $$ 2 a\left(a^{2}+1\right) $$

Write each number in standard notation. See Example 6 Each second, the Sun converts \(7.0 \times 10^{8}\) tons of hydrogen into helium and energy in the form of gamma rays. Write this number in standard notation. (Source: Students for the Exploration and Development of Space)

Square where indicated. Simplify if possible. $$(5 x+2 y)^{2}$$

Multiply. $$(5 x+4)\left(x^{2}-x+4\right)$$

Multiply. $$\left(a^{2}+3 a-2\right)\left(2 a^{2}-5 a-1\right)$$
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