For each trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise. If possible.
`2x^(2) - 7x - 15`

Answer

To determine whether the quadratic expression \(2x^2 - 7x - 15\) is factorable, we will follow these steps:

### Step 1: Identify the coefficients
The quadratic expression can be compared to the standard form \(ax^2 + bx + c\). Here, we have:
- \(a = 2\)
- \(b = -7\)
- \(c = -15\)

### Step 2: Calculate the discriminant \(D\)
The discriminant \(D\) is calculated using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-7)^2 - 4 \cdot 2 \cdot (-15) \]
Calculating this step-by-step:
- First, calculate \((-7)^2 = 49\).
- Next, calculate \(4 \cdot 2 \cdot (-15) = -120\), so \(-4 \cdot 2 \cdot -15 = 120\).
- Now, add these results: \(D = 49 + 120 = 169\).

### Step 3: Check if \(D\) is a perfect square
Since \(D = 169\) and \(169\) is a perfect square (\(13^2\)), the quadratic expression is factorable.

### Step 4: Split the middle term
We need to split the middle term \(-7x\) into two terms whose product is equal to \(a \cdot c\) (which is \(2 \cdot -15 = -30\)) and whose sum is equal to \(-7\).

We need two numbers that multiply to \(-30\) and add up to \(-7\). The numbers are \(3\) and \(-10\):
- \(3 \cdot (-10) = -30\)
- \(3 + (-10) = -7\)

### Step 5: Rewrite the expression
Now we can rewrite the expression \(2x^2 - 7x - 15\) as:
\[ 2x^2 + 3x - 10x - 15 \]

### Step 6: Factor by grouping
Now, we group the terms:
\[ (2x^2 + 3x) + (-10x - 15) \]
Factoring out the common factors in each group:
- From \(2x^2 + 3x\), we can factor out \(x\):
\[ x(2x + 3) \]
- From \(-10x - 15\), we can factor out \(-5\):
\[ -5(2x + 3) \]

Now we can combine these:
\[ x(2x + 3) - 5(2x + 3) \]
Factoring out the common binomial \((2x + 3)\):
\[ (2x + 3)(x - 5) \]

### Final Result
Thus, the factorization of the expression \(2x^2 - 7x - 15\) is:
\[ (2x + 3)(x - 5) \]

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