Q. Simplify the Expression \( (2x – 9)(x + 6) \)

Answer

Use the distributive property.
Multiply each term in the first binomial by each term in the second.

\[ (2x – 9)(x + 6) \]
Expand the expression.
\[ 2x^2 + 12x – 9x – 54 \]
Combine like terms.
12x – 9x = 3x.

\[ 2x^2 + 3x – 54 \]

Detailed Explanation

Simplify the expression

We will simplify the expression by expanding the product step by step and then combining like terms.

Write the original expression:

\( (2x – 9)(x + 6) \)
Use the distributive property (or FOIL) to expand the product.

Distribute each term in the first factor across the second factor:
\[
(2x – 9)(x + 6) = 2x(x + 6) \;-\; 9(x + 6).
\]
Multiply each pair of terms:

Compute the four products (First, Outer, Inner, Last):
- First: \(2x \cdot x = 2x^{2}\)
- Outer: \(2x \cdot 6 = 12x\)
- Inner: \(-9 \cdot x = -9x\)
- Last: \(-9 \cdot 6 = -54\)
So after multiplication we have:
\[
(2x – 9)(x + 6) = 2x^{2} + 12x – 9x – 54.
\]
Combine like terms:

Combine the linear terms \(12x\) and \(-9x\):
\[
12x – 9x = 3x.
\]
Therefore the expression simplifies to:
\[
2x^{2} + 3x – 54.
\]
Final simplified form:

\[
(2x – 9)(x + 6) = 2x^{2} + 3x – 54.
\]

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Frequently Asked Questions

How do you expand (2x - 9)(x + 6)?

Use FOIL or distribution: 2x*x = 2x^2, 2x*6 = 12x, -9*x = -9x, -9*6 = -54. Combine like terms: 2x^2 + 3x - 54.

What is FOIL?

FOIL stands for First, Outer, Inner, Last. It's mnemonic for multiplying two binomials: multiply those four pairs, then combine like terms.

How do I combine like terms correctly?

Add coefficients of the same power: here 12x and -9x give 3x. Keep x^2 and constants separate, so 2x^2 + 3x - 54.

Can I factor 2x^2 + 3x - 54 back to the original factors?

Yes. Factor by grouping or look for two numbers that multiply to -108 and add to 3 (12 and -9), giving (2x - 9)(x + 6).

What are the zeros/roots of the expression?

Set each factor to zero: 2x - 9 = 0 => x = 9/2. x + 6 = 0 => x = -6.

What is the degree and leading coefficient of the simplified polynomial?

Degree is 2 (quadratic). The leading coefficient is 2 (from 2x^2).

How do I evaluate the expression at specific x, say x = 3?

Substitute x = 3 into 2x^2 + 3x - 54: 2(9) + 9 - 54 = 18 + 9 - 54 = -27. Same result from the factored form: (6 - 9)(9) = -27.

What common mistakes should I avoid?

Watch signs (especially negative times positive), multiply every term in each binomial, and only combine like terms (same variable power). Don’t drop terms or mix constants with x-terms.

What does the graph of 2x^2 + 3x - 54 look like?

It's an upward-opening parabol(leading coefficient positive). Vertex x = -b/(2a) = -3/4; y-value is the corresponding minimum. Zeros at x = 9/2 and x = -6.

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