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Evaluate the algebraic expression 3x² - 2xy + 4y² for x=-3 and y=2
Mathematics
Grade 9 (Junior High School)
Question Content
Evaluate $3x^2 - 2xy + 4y^2$ for $x = -3$ and $y = 2$.
Correct Answer
43
Detailed Solution Steps
1
Step 1: Substitute the given values $x=-3$ and $y=2$ into the algebraic expression $3x^2 - 2xy + 4y^2$.
2
Step 2: Calculate each term separately: \n- For $3x^2$: $3\\times(-3)^2 = 3\\times9 = 27$\n- For $-2xy$: $-2\\times(-3)\\times2 = 12$\n- For $4y^2$: $4\\times(2)^2 = 4\\times4 = 16$
3
Step 3: Add the results of the three terms together: $27 + 12 + 16 = 43$
Knowledge Points Involved
1
Substitution in algebraic expressions
This refers to replacing variables in an algebraic expression with their given numerical values. It is the foundational step for evaluating expressions, used whenever you need to find the numerical value of an expression with known variable values.
2
Exponentiation of negative numbers
When a negative number is raised to an even power, the result is positive, e.g., $(-3)^2 = (-3)\\times(-3)=9$. This rule applies to all real numbers and is critical for avoiding sign errors in algebraic evaluation.
3
Multiplication of signed numbers
The product of two negative numbers is positive, and the product of a positive and a negative number is negative, e.g., $-2\\times(-3)\\times2=12$. This rule governs all integer and real number multiplication operations in algebra.
4
Order of operations (PEMDAS/BODMAS)
This rule dictates the sequence of calculations: Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division, and finally Addition and Subtraction. It ensures consistent and correct results when evaluating complex algebraic expressions.
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