Quadratic Formula help?

Question

use the QF to solve the equation 9x^2 - 24x = 3 I cannot get any further than 24 + or - Sqrt 684/18.

use the QF to solve the equation 9x^2 - 24x = 3 I cannot get any further than 24 + or - Sqrt 684/18.

Maybe this will help us more.......
the answers that you can choose from are
12 + sqrt 171 over 9 OR
12 - sqrt 117 over 9 OR
-12 + sqrt 171 over 9 OR
12 - sqrt 171 over 9 OR
12 + sqrt 171 over 18 OR
12 - sqrt 171 over 18 OR
12 + sqrt 117 over 9

select all that apply

Answer 1

When you think about the quadratic formula you think
Ax+By+C=0 form

You have 9x^2 - 24x = 3 that would mean 9x^2 - 24x -3 = 0
So
A=9
B=-24
C=-3

Then if you already know the quadratic equation you have to thouroghly check each symbol you put in.

Just plug in the numbers part by part

(b^2 - 4ac) = (24^2 - 4(9)(-3)) = (576 - 108) = 468

Then on to part 2

-b+or- sqrt(468) = 24+or- sqrt(468) ((BUT WAIT !))

468 cant be squarerooted evenly the best way to do this is to factor out the radical.

24+or- sqrt(468) = (24 + or - 6√13)

Ok on to part 3 the final part

You now have (24 ± 6√13) / 2a = (24 ± 6√13) /18

Now simplify (24 ± 6√13) /18 = 4/3+√13/3

So you cant go any further the answer is:

4/3 ± (√13)/3

Answer 2

Quadratic formula: x = [- b +/- V`(b^2 - 4ac)] / 2a
>>> (V` represents the square root/radical sign).

First: set the equation to "0" > subtract 3 from both sides and
determine a, b, and c >

9x^2 - 24x - 3 = 3 - 3
9x^2 - 24x - 3 = 30 > (a = 9, b = -24, c = 3)

*Replace the coefficients with the appropriate variable in the formula >

x = [- (-24) +/- V`((-24)^2 - 4(9)(3)] / 2(9)
x = [24 +/- V`(576 - 36(3))] / 18
x = [24 +/- V`(576 - 108)] / 18
x = [24 +/- V`(468)] / 18

Second: simplify 468 into lowest terms >

x = [24 +/- V`(9*52)] / 18
x = [24 +/- V`(3*3*4*13)] / 18
x = [24 +/- V`(3*3*2*2*13)] / 18
x = [24 +/- 3*2 V`13] / 18
x = [24 +/- 6 V`13] / 18

Third: solve for "x" using addition and subtraction separtely >

1. x = [24 + 6 V`13] / 18
x = 24/18 + (6 V`13)/18
x = 4/3 + (1 V`13)/3

2. x = [24 - 6 V`13] / 18
x = 24/18 - (6 V`13)/18
x = 4/3 - (1 V`13)/3

Answer 3

ok the other people who have answered so far with

4/3 + sqrt(19)/3 and 4/3 - sqrt(19)/3 are correct.

We just need to work this out algebraically to get the answer choices.

4/3 is the samething thing as 12/9. To change the denominator to 9 on the right term, multiply top and bottom of the fraction by 3.

sqrt(19) * 3 / (3 * 3) = 3 * sqrt(19) / 9

now we need to bring the 3 into the square root. To do this, square the 3 and then multiply it by the 19. 3^2 = 9; 19 * 9 = 171.

so the answers are

( 12 + sqrt(171) ) / 9
( 12 - sqrt(171) ) / 9

Answer 4

First you have to make sure that the equation is set equal to 0.
So, 9x^2-24x-3=0

Quadratic Formula is x= (-b ± sqrt(b^2-4ac))/2a
a=9, b=-24, c=-3

x = (24 ±sqrt(24^2-4*9*-3))/2*9

x = (24± sqrt(576+108))/18

x = (24± sqrt(684))/18 <---- This is where you are stuck.

Now let's look at sqrt(684).... 684 is 19*36...so this would be 6|19 (using | to mean sqrt)...

x = (24 ± 6|19)/18

x = 24/18 ± 6|19/18

x = 4/3 ± |19/3

Answer 5

You can simplify sqrt(684) down to 6*sqrt(19) and then simplify the entire expression down to ( 4+ or - sqrt(19))/3

Answer 6

9x^2 - 24x - 3 = 0
<=> 3x^2 - 8x -1 = 0
delta = 8^2 + 4*3 = 76
=> sqrt(delta) = 2sqrt(19)
x1 = ( 8 - 2sqrt(19) ) / 6 = ( 4 - sqrt(19) ) / 3
x2 = ( 8 + 2sqrt(19)) / 6 = ( 4 + sqrt(19) ) / 3

Answer 7

The best that you can do now is to simplify your sqrt to 6sqrt19. Then, you can simplifythe whole equation by dividing everything by six. Your final answer should be

(4+or-(sqrt(19)))/3

I hope that this helps.

Answer 8

9x^2 - 24x = 3
9x^2 - 24x - 3 = 0
3(3x^2 - 8x - 1) = 0
3x^2 - 8x - 1 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-(-8) ± sqrt((-8)^2 - 4(3)(-1)))/(2(3))
x = (8 ± sqrt(64 + 12))/6
x = (8 ± sqrt(76))/6
x = (8 ± sqrt(4 * 19))/6
x = (8 ± 2sqrt(19))/6
x = (1/3)(4 ± sqrt(19))

Answer 9

9x^2 - 24x = 3

Divide by 3 and collect all terms in one side,
3x^2-8x-1=0

By quadratic formula,
x = (8±√(8^2+12)/6 = 4/3 ± √(19) / 3