Q. \(x^2 – 10x + 24\)

Problem: Solve the expression \(x^2 – 10x + 24\). We will factor it to find its roots.

Step 1: Identify the form

The expression \(x^2 – 10x + 24\) is a quadratic polynomial of the form

\[ax^2 + bx + c\]

Here, \(a = 1\), \(b = -10\), and \(c = 24\).

Step 2: Factor using the product and sum

We want to write the quadratic as a product of two binomials:

\[(x – m)(x – n)\]

When expanded, this becomes

\[x^2 – (m + n)x + mn\]

We match coefficients with \(x^2 – 10x + 24\).

Step 3: Match the constant term

The constant term in the factored form is \(mn\). We need

\[mn = 24\]

Step 4: Match the middle coefficient

The coefficient of \(x\) in the factored form is \(-(m + n)\). We need

\[-(m + n) = -10\]

So we require

\[m + n = 10\]

Step 5: Find numbers with product 24 and sum 10

List factor pairs of \(24\). The pair that adds to \(10\) is \(6\) and \(4\).

\[6 \cdot 4 = 24,\quad 6 + 4 = 10\]

Step 6: Write the factored form

Substitute \(m = 6\) and \(n = 4\) into \((x – m)(x – n)\):

\[(x – 6)(x – 4)\]

Final Answer

So the expression factors as

\[x^2 – 10x + 24 = (x – 6)(x – 4)\]