Question

1. determine the factors of the following quadratic equation: $3x^2 + x - 10 = 0$. teks a2.4(f)

a) $(5x - 2)(x - 2)$
b) $(2x + 1)(x - 2)$
c) $(3x - 5)(x - 2)$
d) $(3x - 5)(x + 2)$

Explanation:

Step1: Expand Option A

Multiply \((5x - 2)(x - 2)\) using the distributive property (FOIL method):
\[

$$\begin{align*} (5x - 2)(x - 2)&=5x\cdot x+5x\cdot(-2)-2\cdot x - 2\cdot(-2)\\ &=5x^{2}-10x - 2x + 4\\ &=5x^{2}-12x + 4 \end{align*}$$

\]
This does not match \(3x^{2}+x - 10\), so A is incorrect.

Step2: Expand Option B

Multiply \((2x + 1)(x - 2)\):
\[

$$\begin{align*} (2x + 1)(x - 2)&=2x\cdot x+2x\cdot(-2)+1\cdot x+1\cdot(-2)\\ &=2x^{2}-4x+x - 2\\ &=2x^{2}-3x - 2 \end{align*}$$

\]
This does not match \(3x^{2}+x - 10\), so B is incorrect.

Step3: Expand Option C

Multiply \((3x - 5)(x - 2)\):
\[

$$\begin{align*} (3x - 5)(x - 2)&=3x\cdot x+3x\cdot(-2)-5\cdot x - 5\cdot(-2)\\ &=3x^{2}-6x - 5x + 10\\ &=3x^{2}-11x + 10 \end{align*}$$

\]
This does not match \(3x^{2}+x - 10\), so C is incorrect.

Step4: Expand Option D

Multiply \((3x - 5)(x + 2)\):
\[

$$\begin{align*} (3x - 5)(x + 2)&=3x\cdot x+3x\cdot2-5\cdot x - 5\cdot2\\ &=3x^{2}+6x - 5x - 10\\ &=3x^{2}+x - 10 \end{align*}$$

\]
This matches the given quadratic equation.

Answer:

D) \((3x - 5)(x + 2)\)