Factories `x ^(2) + 3x - 18` (a) (x+18)(x-1) (b) (x-1)(c+18) (c) (x+6)(x-3) (d) (x-6)(x+3)

Options

A. `(x + 18) (x-1)`
B. `(x-1) (c + 18)`
C. `(+6) (-3)`
D. `(x-6) (x +3)`

Answer

To factor the quadratic expression \(x^2 + 3x - 18\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is \(x^2 + 3x - 18\). Here, the coefficients are:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 3\) (coefficient of \(x\))
- \(c = -18\) (constant term)

### Step 2: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(a \cdot c = 1 \cdot (-18) = -18\) and add to \(b = 3\).

The pairs of factors of \(-18\) are:
- \(1\) and \(-18\)
- \(-1\) and \(18\)
- \(2\) and \(-9\)
- \(-2\) and \(9\)
- \(3\) and \(-6\)
- \(-3\) and \(6\)

Among these, the pair that adds up to \(3\) is \(6\) and \(-3\).

### Step 3: Rewrite the middle term using the found numbers
We can rewrite the expression \(x^2 + 3x - 18\) as:
\[ x^2 + 6x - 3x - 18 \]

### Step 4: Factor by grouping
Now, we will group the terms:
\[ (x^2 + 6x) + (-3x - 18) \]

Next, we factor out the common terms in each group:
\[ x(x + 6) - 3(x + 6) \]

### Step 5: Factor out the common binomial
Now, we can factor out the common binomial \((x + 6)\):
\[ (x + 6)(x - 3) \]

### Conclusion
Thus, the factorization of \(x^2 + 3x - 18\) is:
\[ (x + 6)(x - 3) \]

The correct option from the given choices is (c) \((x + 6)(x - 3)\).

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