Solving Linear/ Quadratic Equation: Solve y=9x-4 and y=2x^2

[MathsFormula]

Solve y=9x-4 and y=2x²

I get stuck on the last stage:

9x - 4=2x²

2x² - 9x + 4 = 0

I need to find two numbers that multiply to give 8 and add up to give -9 (the 8 is from the 2 and the 4 in the Quadratic Equation. The -9 is in the Equation too).

The numbers are -8 and -1

The '-9×' term is split to give :

2x² - 8x - 1x + 4 = 0

x(2x-8) - (x - 4) = 0

Now (x-4) is the common term but what do I do now???

Answer is x=1/2 y=1/2 or x=4 y=32

Please help. Thank you

 

[ksdhart2]

You're almost there. To use a metaphor, you just stopped running about three inches shy of the goal line. You've correctly identified (x - 4) as the common factor, so continue by factoring it out.

\(\displaystyle x(2x-8)-(x-4)=0 \implies (x-4)(2x-1)=0\)

Now, what do you know about the values of two things that multiply to 0? What does that suggest the two values of x are? Plugging them into the given formula, what does that make the values of y? Do those match the answers you were given?

 

[MathsFormula]

\(\displaystyle x(2x-8)-(x-4)=0 \implies (x-4)(2x-1)=0\)

I can see that now thanks BUT what are the lines in between or thinking that got you there?

 

[MathsFormula]

Whoa!
Can you solve that equation using the quadratic formula?

Forget the variable y: of no importance...

Using the Quadratic formula I got x= 4 and x= 1.

X=1 is WRONG

 

[HallsofIvy]

"In between" factoring (which often involves a little "guess and check") and the quadratic equation (which is often applied without any thinking at all) is "completing the square".

For any x and a, \(\displaystyle (x- a)^2= x^2- 2ax+ a^2\). Comparing \(\displaystyle 2x^2 - 9x= 2(x^2+ (9/2)z\) with that we see that 2a= 9/2 and a= 9/4 so that \(\displaystyle a^2= 81/16\). So we can write \(\displaystyle 2x^2- 9x+ 4= 2(x^2- (9/2)x+ 81/4- 81/4)+ 4= 2(x- 9/4)^2- 81/2+ 4= 2(x^2- 9/4)^2- 73/4= 2(x- 9/4- \sqrt{73}{2})(x- 9/4- \sqrt{73}{2})\).

 

[MathsFormula]

Try again: 4 correct, x should be 1/2

Thanks so far. I got the answer using the quadratic formula but still don't understand the factorising method

 

[Steven G]

I can see that now thanks BUT what are the lines in between or thinking that got you there?

No, you did not see it now. If you did you would not be asking BUT what are the lines in between. Not a big deal as I will show you the missing steps.


x(2x-8) - (x-4) = 0
2x(x-4) - 1(x-4) = 0 (I just factored out the 2 from (2x-8)

I now will factor out what is in common (in red) and I will put what is left over (blue) in parenthesis.

2x(x-4) - 1(x-4) = 0

(x-4)(2x-1) = 0 Continue from here.

 

[MathsFormula]

No, you did not see it now. If you did you would not be asking BUT what are the lines in between

QUITE RIGHT. Now I understand. Thanks