Let's factorise the quadratic expression [tex]\(10x^2 - 9x + 2\)[/tex].
1. Identify the quadratic expression:
[tex]\[
10x^2 - 9x + 2
\][/tex]
2. Look for two binomials that multiply to give the quadratic expression. These binomials will typically take the form:
[tex]\[
(ax + b)(cx + d)
\][/tex]
where [tex]\(a \cdot c = 10\)[/tex] (the coefficient of [tex]\(x^2\)[/tex]), and [tex]\(b \cdot d = 2\)[/tex] (the constant term).
3. We also need [tex]\(a \cdot d + b \cdot c\)[/tex] to equal [tex]\(-9\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
After looking at the coefficients, let’s consider possible pairs of factors:
- For the coefficient of [tex]\(x^2\)[/tex] ([tex]\(10\)[/tex]): possible pairs are [tex]\((10, 1)\)[/tex] and [tex]\((5, 2)\)[/tex].
- For the constant term ([tex]\(2\)[/tex]): possible pairs are [tex]\((2, 1)\)[/tex] and [tex]\((-2, -1)\)[/tex].
We need to find the right combination such that the middle term adds up to [tex]\(-9x\)[/tex].
After trying different combinations, we identify the correct pairings as:
[tex]\[
(2x - 1) \text{ and } (5x - 2)
\][/tex]
4. Verify the factorisation:
[tex]\[
(2x - 1)(5x - 2)
\][/tex]
Expand it to check:
[tex]\[
(2x - 1)(5x - 2) = 2x \cdot 5x + 2x \cdot (-2) + (-1) \cdot 5x + (-1) \cdot (-2)
\][/tex]
[tex]\[
= 10x^2 - 4x - 5x + 2
\][/tex]
[tex]\[
= 10x^2 - 9x + 2
\][/tex]
Thus, the correct factorisation of [tex]\(10x^2 - 9x + 2\)[/tex] is:
[tex]\[
(2x - 1)(5x - 2)
\][/tex]
