Question

How do you factor 3x^2 - 10 + 3 = 0?

Answer 1

Answer:

3 and 1/3

Step-by-step explanation:

3x² - 10 + 3 = 0

To factorise, we need two numbers whose sum is 10 and their product is 9.

So, two such numbers are 9 and 1.

On splitting it's middle term,

3x² - 10x + 3 = 0

⇒ 3x² - 9x - x + 3 = 0

⇒ 3x ( x - 3 ) - 1 ( x - 3 ) = 0

⇒ ( 3x - 1 ) ( x - 3 ) = 0

By zero product rule ;

x = 1/3 or x = 3

Hence, the zeroes are 1/3 and 3.

Answer 2

Factorsing 3x² - 10x + 3 = 0

As the polynomial is equal to 0, this means that, we'll get any two solutions of it, so that, after substituting it in theplace of x, we'll get the answer as 0.

We can factorise it by using various methods, but for our ease of understanding I'll do it by usin' splitting the middle term method.

Step 1: Check the polynomial. If it's not in its general form i.e., ax² + bx + c rearrange the polynomial according to the general form. But don't change its signs!

Here, the polynomial 3x² - 10x + 3 is in its general form so we don't have to rearrange it.

Step 2: Multiplying the co - efficient of x² and the constant term.

Here, the co-efficient of x² is 3 and the constant term is 3.

Multiplying 3 and 3, we obtain, 9.

Step 3: We've to find the two factors of 9 which would give us the sum as the co - efficient of x.

Here, the co-efficient of x is - 10.

So, the factors of 9 are $\bold{\mathsf{\pm 1, \pm 3, \pm 9}}$

The two factors of 9 which would give us the sum as - 10 are - 1 and - 9.

Step 4: Substituting - 1 and - 9 in the place of - 10 along with the respective variable. Your can write any of them first.

So, 3x² - 10x + 3

=> 3x² - x - 9x + 3 (or)

3x² - 9x - x + 3

Step 5: Now group the terms.

(3x² - x) - (9x + 3)

Step 6: Take common terms outside.

x(3x - 1) - 3(3x - 1)

Step 7: You'll observe that, both the terms inside one bracket will be the same as other. Take common outside and uncommon terms inside the bracket to be written after the common terms.

(3x - 1)(x - 3)

Therefore, 3x - 1 and x - 3 are the factors of 3x^2 - 10x + 3.

But 3x² - 10x + 3 = 0

So, (3x - 1) and (x - 3) should be equal to 0.

Step 8: Equating (3x - 1) and (x - 3) with 0.

(3x - 1)(x - 3) = 0

(3x - 1) = 0

3x - 1 = 0

3x = 1

$\fbox{\bold{\mathsf{x = \frac{1}{3}}}}$

(x - 3) = 0

x - 3 = 0

$\fbox{\bold{\mathsf{x = 3}}}$

Therefore, x = $\frac{1}{3}$ or x = 3.

$\fbox{\bold{\mathsf{Note:}}}$

If you are just factoring the polynomial, then step 8 is not needed as it's about how to find the value of x.

The quadratic polynomial always has, no zero, one zero, maximum two zeros and not more than that.