In the algebraic expression 
5
(
4
x
2
+
q
x
+
9
)
−
15
x
(
q
+
5
)
5(4x 
2
 +qx+9)−15x(q+5), where 
q
q is a constant, it is said to be equivalent to 
20
x
2
−
180
x
+
45
20x 
2
 −180x+45. What is the constant 
q
q?

To find the constant \( q \) in the given expression, we'll start by expanding the expression and then comparing it with the given equivalent expression.

Given:
\[
5(4x^2 + qx + 9) - 15x(q + 5)
\]
is equivalent to:
\[
20x^2 - 180x + 45
\]

### Step 1: Expand the given expression
First, distribute the 5 in the first term:
\[
5(4x^2 + qx + 9) = 20x^2 + 5qx + 45
\]

Next, distribute the \(-15x\) in the second term:
\[
-15x(q + 5) = -15qx - 75x
\]

Now, combine both parts:
\[
20x^2 + 5qx + 45 - 15qx - 75x
\]

### Step 2: Combine like terms
Combine the terms with \( x \):
\[
20x^2 + (5q - 15q)x + 45 - 75x
\]
Simplifying further:
\[
20x^2 + (5q - 15q - 75)x + 45
\]
This simplifies to:
\[
20x^2 + (5q - 15q - 75)x + 45
\]
\[
20x^2 + (-10q - 75)x + 45
\]

### Step 3: Compare with the given equivalent expression
The given expression is:
\[
20x^2 - 180x + 45
\]

Now, by comparing the coefficients of the terms involving \( x \), we have:
\[
-10q - 75 = -180
\]

### Step 4: Solve for \( q \)
To solve for \( q \), set up the equation:
\[
-10q - 75 = -180
\]
Add 75 to both sides:
\[
-10q = -105
\]
Divide by \(-10\):
\[
q = 10.5
\]

So, the constant \( q \) is \( 10.5 \).

---

Would you like more details or have any questions? 

Here are 8 related questions to consider:
1. What are the properties of polynomials and their degrees?
2. How do you factorize a quadratic expression?
3. How do you solve equations involving multiple variables?
4. What is the importance of comparing coefficients in algebraic expressions?
5. How can you use algebra to solve word problems?
6. What are some common mistakes when expanding and simplifying expressions?
7. How can substitution be used in solving algebraic expressions?
8. What methods are there to verify the solutions of algebraic equations?

**Tip:** Always double-check your algebraic manipulations by substituting your answer back into the original equation.

Learn how to determine the constant q in the algebraic expression 5(4x^2 + qx + 9) - 15x(q + 5), which is equivalent to 20x^2 - 180x + 45. This problem involves expanding algebraic expressions, comparing coefficients, and solving equations. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation