#### C. SOLUTION OF EQUATIONS

Question: **Solve** 3x^{3} – 10x^{2} + 9x – 2 **= 0**

When you spot the keywords

– “Solve”and

– equation = 0

– “Find the roots of equation”in your exam question, it means you’ll have to:

- Find the factors of an equation f(x) (usually cubic)
- Find the
*roots*of f(x)=0 (i.e. final answer must be in the form: x = a, b … where a, b, … are the roots)

Ans: Let f(x) = 3x^{3} – 10x^{2} + 9x – 2

Find the first factor via trial and error

Try x=1: f(1) = 3(1)

^{3}– 10(1)^{2}+ 9(1) – 2 = 0

⇒ (x-1) is a factorFind the remainder expression by either LONG DIVISION or COMPARING COEFFICIENTs

## Comparing Coefficient Method

Form the expression after factorising f(x) with the 1st factor found

3x^{3}– 10x^{2}+9x-2 ≡ (x-1)(ax^{2}+bx+c)

Compare coefficients of highest power and constant first

Compare x^{3}:

3x^{3 }= ax^{3} –> a=3

Compare constant:

-2= -c –> c=2

Then compare coefficients of middle powers

Compare x

9x=cx-bx

Substitute c=2

9x=2x-bx

b=-7

so f(x)= (x-1)(3x^{2}-7x+2) 🙂

But Question wants us to find all the roots, solve for all x, so we have to further solve (3x

^{2}-7x+2) , there are 4 methods for you to choose from, but i will use the frame method.

f(x)=(x-1)(x-2)(3x-1)

so x-1=0, x-2=0, 3x-1=0

so x=1 or 2 or 1/3 🙂

Differentiate between question asking you to

- Factorize –> f(x)=(x-1)(x-2)(3x-1)
- Solve –> x=1 or 2 or 1/3 🙂

If the quadratic equation in Step 3 cannot be easily factorized → you’ll have to use the method 4 in Quadratic Formula to find the two solutions. You’ll normally need to use method 4 when you see terms like ± or

√surdswithin the question or when the question ask you to leave the answer inexact form.

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