Quadratic equation, cannot solve it while using a technique that doesn't use bhaskara

Question

My teacher taught how to solve squared equations without bhaskara. It's a completely new technique for me. Actually I solve only 2 problems till now. I am having a hard time solving the last one of the exercise list. Below I will show where I am stuck. Excuse me if I use terms that are not right, I am not certain how to say them in english.

14X² -43x + 20 = 0 because we have (+20) the two factors will have the same sign.

bellow I am turning 20 into products

14X² -43x + 20 = 0
            1 - 20
            2 - 10
            4 - 5

now I turng 14X² into products

14X² -43x + 20 = 0
x - 14x       1 - 20
2x - 7x       2 - 10
              4 - 5

Now I need to multiply the factors of x and +20 and sum them to get -43. In other words I need to find the maching pairs. But there's not matching pairs. Apparently. What I am doing wrong?

Answer 1

$$14x^2-35x-8x+20=0$$ $$\implies 7x(2x-5)-4(2x-5)=0$$ $$\implies(7x-4)(2x-5)=0$$

Now you can easily find the roots.

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(No need to read further if you are good with the middle term factorization)

Some more tips about using the technique - Middle term Factorization

In India, we are taught this method before Bhaskara. Some points that may help you in splitting up easily

1. Make the coefficient of $x^2$ positive. In most of the cases you have a positive coefficient, but if they give you a negative coefficient, then you can multiply both sides by $-1$. Now the equation would look like $ax^2±bx+c$ or $ax^2±bx-c$ where $a$, $b$ and $c$ are positive numbers
2. Prime factorize $a×c$.
3. Now, if the sign of $c$ is positive, then the splitting is to be done by addition. If the sign of $c$ is negative, then it is to be done by subtraction.
4. Combine all the factors in two numbers in such a way that they either add up or their difference is $b$ (the addition or subtraction is to be decided by point 3.)

For instance, $$14x^2-43x+20$$

Step 1 - Already done

Step 2 - $14×35=2×2×2×5×7$ . So the factors are $2,2,2,5,7$

Step 3 - As the sign of $c$ is positive, so the splitting is to be done by addition.

Step 4 - This is to be totally done by hit-and-trial. However the third point helps us that both the factors must be less than 43. We have to only check by addition, etc. It can be easily spotted that $43=35+8$

Another example: $$7\sqrt2x^2+10x-4\sqrt2$$

Step 1 - Done

Step 2 - $7\sqrt2×4\sqrt2=2×2×2×7$

Step 3 - Factorization to be done by subtraction. So one of the numbers must be greater than $10$

Step 4 - $14-4=10$