For each trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise. If possible.
`x(2x-1)-1`

Answer

To determine whether the given expression \( x(2x-1) - 1 \) is factorable, we will follow these steps:

### Step 1: Expand the expression
We start by expanding the expression \( x(2x - 1) - 1 \).

\[ x(2x - 1) - 1 = 2x^2 - x - 1 \]

### Step 2: Identify coefficients
Now, we identify the coefficients \( a \), \( b \), and \( c \) from the quadratic expression \( 2x^2 - x - 1 \).

- \( a = 2 \)
- \( b = -1 \)
- \( c = -1 \)

### Step 3: Calculate the discriminant
Next, we calculate the discriminant \( D \) using the formula \( D = b^2 - 4ac \).

\[ D = (-1)^2 - 4 \cdot 2 \cdot (-1) = 1 + 8 = 9 \]

### Step 4: Check if the discriminant is a perfect square
Since \( D = 9 \) is a perfect square, the quadratic expression is factorable.

### Step 5: Factor the quadratic expression
To factor \( 2x^2 - x - 1 \), we look for two numbers that multiply to \( a \cdot c = 2 \cdot (-1) = -2 \) and add to \( b = -1 \). The numbers are \( 1 \) and \( -2 \).

Now we can rewrite the middle term:

\[ 2x^2 - 2x + x - 1 \]

### Step 6: Group the terms
Next, we group the terms:

\[ (2x^2 - 2x) + (x - 1) \]

### Step 7: Factor by grouping
Now we factor out the common factors from each group:

\[ 2x(x - 1) + 1(x - 1) \]

### Step 8: Factor out the common binomial
We can now factor out the common binomial \( (x - 1) \):

\[ (2x + 1)(x - 1) \]

### Final Answer
Thus, the expression \( x(2x - 1) - 1 \) factors to:

\[ (2x + 1)(x - 1) \]