Factorise:
`15x ^(2)- 26 x + 8`

Answer

To factorise the quadratic expression \(15x^2 - 26x + 8\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 15\)
- \(b = -26\)
- \(c = 8\)

### Step 2: Calculate the product \(ac\)
We need to find the product of \(a\) and \(c\):
\[ ac = 15 \times 8 = 120 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(120\) (the value of \(ac\)) and add up to \(-26\) (the value of \(b\)).

The two numbers that satisfy these conditions are \(-20\) and \(-6\) because:
\[ -20 \times -6 = 120 \quad \text{and} \quad -20 + (-6) = -26 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression by splitting the middle term using the two numbers we found:
\[ 15x^2 - 20x - 6x + 8 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (15x^2 - 20x) + (-6x + 8) \]

### Step 6: Factor out the common factors from each group
From the first group \(15x^2 - 20x\), we can factor out \(5x\):
\[ 5x(3x - 4) \]
From the second group \(-6x + 8\), we can factor out \(-2\):
\[ -2(3x - 4) \]

### Step 7: Combine the factors
Now we can combine the factored groups:
\[ 5x(3x - 4) - 2(3x - 4) \]
This can be written as:
\[ (3x - 4)(5x - 2) \]

### Final Answer
Thus, the factorised form of \(15x^2 - 26x + 8\) is:
\[ (3x - 4)(5x - 2) \]

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