Question
what other expressions simplify to the same thing as 4x^2 -3x^2+5x-24.
Answers
GPT-4o mini
To find other expressions that simplify to the same thing as \( 4x^2 - 3x^2 + 5x - 24 \), we first simplify the original expression:
\[
4x^2 - 3x^2 + 5x - 24 = (4 - 3)x^2 + 5x - 24 = 1x^2 + 5x - 24 = x^2 + 5x - 24
\]
Now that we have simplified the expression to \( x^2 + 5x - 24 \), we can create equivalent expressions by performing operations such as factoring, adding zero, or multiplying by one.
1. **Factoring**:
The expression can potentially be factored. To factor \( x^2 + 5x - 24 \), we look for two numbers that multiply to \(-24\) and add to \(5\). The factors \(8\) and \(-3\) work:
\[
x^2 + 5x - 24 = (x + 8)(x - 3)
\]
2. **Adding zero**:
We can add and subtract the same term without changing the value:
\[
x^2 + 5x - 24 + 0 = x^2 + 5x - 24 + (2 - 2) = x^2 + 5x - 24 + 2 - 2 = x^2 + 5x - 22 - 2
\]
3. **Multiplying by one**:
We can also multiply the expression by \(1\), which could be expressed in various forms, for instance:
\[
x^2 + 5x - 24 = 1(x^2 + 5x - 24) = 2 \cdot \frac{1}{2} (x^2 + 5x - 24)
\]
4. **Combining like terms or rearranging**:
As long as we maintain the balance of the equation, we can rearrange terms:
\[
x^2 - 24 + 5x = 5x + x^2 - 24
\]
Therefore, here are examples of expressions that simplify to the same value as \( x^2 + 5x - 24 \):
- \( (x + 8)(x - 3) \)
- \( x^2 + 5x - 22 - 2 \)
- \( 2 \cdot \frac{1}{2} (x^2 + 5x - 24) \)
- \( 5x + x^2 - 24 \)
All of these expressions are equivalent to the original one.
\[
4x^2 - 3x^2 + 5x - 24 = (4 - 3)x^2 + 5x - 24 = 1x^2 + 5x - 24 = x^2 + 5x - 24
\]
Now that we have simplified the expression to \( x^2 + 5x - 24 \), we can create equivalent expressions by performing operations such as factoring, adding zero, or multiplying by one.
1. **Factoring**:
The expression can potentially be factored. To factor \( x^2 + 5x - 24 \), we look for two numbers that multiply to \(-24\) and add to \(5\). The factors \(8\) and \(-3\) work:
\[
x^2 + 5x - 24 = (x + 8)(x - 3)
\]
2. **Adding zero**:
We can add and subtract the same term without changing the value:
\[
x^2 + 5x - 24 + 0 = x^2 + 5x - 24 + (2 - 2) = x^2 + 5x - 24 + 2 - 2 = x^2 + 5x - 22 - 2
\]
3. **Multiplying by one**:
We can also multiply the expression by \(1\), which could be expressed in various forms, for instance:
\[
x^2 + 5x - 24 = 1(x^2 + 5x - 24) = 2 \cdot \frac{1}{2} (x^2 + 5x - 24)
\]
4. **Combining like terms or rearranging**:
As long as we maintain the balance of the equation, we can rearrange terms:
\[
x^2 - 24 + 5x = 5x + x^2 - 24
\]
Therefore, here are examples of expressions that simplify to the same value as \( x^2 + 5x - 24 \):
- \( (x + 8)(x - 3) \)
- \( x^2 + 5x - 22 - 2 \)
- \( 2 \cdot \frac{1}{2} (x^2 + 5x - 24) \)
- \( 5x + x^2 - 24 \)
All of these expressions are equivalent to the original one.