For each trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise. If possible.
`2x^(2) + 2x - 75`

Answer

To determine whether the quadratic expression \(2x^2 + 2x - 75\) is factorable, we will follow these steps:

### Step 1: Identify coefficients
The given expression is in the form \(ax^2 + bx + c\). Here, we identify:
- \(a = 2\)
- \(b = 2\)
- \(c = -75\)

### 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 \(c\):
\[ D = 2^2 - 4 \cdot 2 \cdot (-75) \]

### Step 3: Simplify the expression for D
Calculating \(D\):
\[ D = 4 - 4 \cdot 2 \cdot (-75) \]
Calculating \(4 \cdot 2 \cdot (-75)\):
\[ 4 \cdot 2 = 8 \]
\[ 8 \cdot (-75) = -600 \]
Thus:
\[ D = 4 - (-600) = 4 + 600 = 604 \]

### Step 4: Check if D is a perfect square
Now, we need to check if \(D = 604\) is a perfect square. The square root of \(604\) is approximately \(24.6\), which is not an integer. Therefore, \(604\) is not a perfect square.

### Step 5: Conclusion
Since the discriminant \(D\) is not a perfect square, the quadratic expression \(2x^2 + 2x - 75\) is not factorable over the integers.

### Final Answer
The expression \(2x^2 + 2x - 75\) is not factorable.

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