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Chapter 3: Problem 6

Evaluate the expression at the given value of \(\mathrm{x}\). $$4 x^{2}-9 x+4\( at \)x=-3$$

Short Answer

Expert verified
The expression evaluates to 67 when \(x = -3\).

Step by step solution

01

Substitute the Value of x

We need to evaluate the expression \(4x^2 - 9x + 4\) at \(x = -3\). Substitute \(x = -3\) into the expression: \(4(-3)^2 - 9(-3) + 4\).
02

Evaluate the Squares and Products

Calculate \((-3)^2\), which equals 9. Then multiply by 4 to get \(4 imes 9 = 36\). Next, compute \(-9 imes (-3) = 27\). Substitute back into the expression to get \(36 + 27 + 4\).
03

Combine the Terms

Add the resulting numbers from the previous step: \(36 + 27 = 63\). Then add 4: \(63 + 4 = 67\). This gives us the final evaluation of the expression.

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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 technique used in algebra to replace variables with known values. In essence, it allows us to solve equations or expressions where we are given specific values for variables. When we substitute, we essentially "plug in" these given values into every instance of the variable in the expression.
In the original exercise, we were given the polynomial expression \(4x^2 - 9x + 4\) and needed to evaluate it at \(x = -3\). Here's how substitution works in practice:
  • Identify the variable in the expression that needs substitution鈥攊n this case, it's \(x\).
  • Substitute \(x\) with \(-3\), transforming the expression into \(4(-3)^2 - 9(-3) + 4\).
  • Once substituted, the expression can be evaluated numerically.
This step simplifies the problem by reducing it into a straightforward calculation. It's an essential step to make complex expressions more manageable.
Order of Operations
The order of operations sets rules on which computations to perform first in mathematical expressions. It鈥檚 often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Following these rules ensures that calculations are performed consistently and correctly.
  • First, solve any operations in parentheses.
  • Then, address the exponents. For \((-3)^2\), compute the square to get 9.
  • Next, handle multiplication: Multiply 4 by 9 to yield 36, and multiply \(-9\) by \(-3\) to get 27.
  • Finally, perform addition and subtraction in the expression. So, add together 36, 27, and 4.
Remembering the correct order is crucial, as performing operations out of sequence can lead to incorrect results. Moving step by step ensures each part of the expression is correctly addressed.
Polynomial Evaluation
Polynomial evaluation involves calculating the value of a polynomial using given variable values. Polynomials are expressions that contain variables raised to various powers and coefficients.
This is the process we'll follow:
  • Identify the polynomial structure: In this exercise, we worked with \(4x^2 - 9x + 4\).
  • Substitute the specified value (\(x = -3\)) into the polynomial.
  • Evaluate the expression, taking care of each term as per the order of operations.
Evaluating polynomials is a foundational skill in algebra, helping to simplify expressions and solve equations effectively. By understanding each component of a polynomial and correctly applying operations, you can accurately determine its value for any given variable.

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