Math Problem Statement

已知关于x的二次多项式a(x^2 - x + 3x) + b(x^2 + x) + x^2 - 5, 当x = -2时,该多项式的值为多少?

Solution

The image contains a math problem, and it appears to be in Chinese. From what I can make out, it seems to involve a polynomial expression. I'll break down the problem and solve it based on the visible content.

The equation given is:

a(x2x+3x)+b(x2+x)+x25whenx=2a(x^2 - x + 3x) + b(x^2 + x) + x^2 - 5 \quad \text{when} \quad x = -2

You are asked to find the value of this expression when x=2x = -2, and there is some missing context about values for aa and bb.

Let's substitute x=2x = -2 and simplify the expression, assuming the coefficients aa and bb are not yet given.

  1. Simplify each term inside the parentheses:
    • x2x+3x=x2+2xx^2 - x + 3x = x^2 + 2x
    • x2+xx^2 + x

So the expression becomes:

a(x2+2x)+b(x2+x)+x25a(x^2 + 2x) + b(x^2 + x) + x^2 - 5

Now, substituting x=2x = -2:

a((2)2+2(2))+b((2)2+(2))+(2)25a((-2)^2 + 2(-2)) + b((-2)^2 + (-2)) + (-2)^2 - 5

  1. Calculate the terms:
    • (2)2=4(-2)^2 = 4
    • 2(2)=42(-2) = -4
    • (2)2=4(-2)^2 = 4
    • 2=2-2 = -2
    • (2)25=45=1(-2)^2 - 5 = 4 - 5 = -1

Now, the expression becomes:

a(44)+b(42)+45=0a+2b1a(4 - 4) + b(4 - 2) + 4 - 5 = 0a + 2b - 1

This simplifies to:

2b12b - 1

So, the value of the expression when x=2x = -2 is 2b12b - 1.

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Math Problem Analysis

Mathematical Concepts

Polynomial Expressions
Substitution
Simplification

Formulas

Substitution into polynomial expressions

Theorems

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Suitable Grade Level

Grades 9-11