how do you solve these??

Question

4x^2 - 81 = 0

x^2 - 2x - 24 = 0

3x^2 - x - 4 = 0

Answer 1

First one...
4x^2 - 81 = 0

The goal is to get x by itself... So start with adding "81" to both sides to get it to cancel out on the left side...
4x^2 - 81 + (81) = + (81)

Simplify...
4x^2 = 81

Divide by "4" on both sides to get it to cancel on the left...
4x^2 / (4) = 81 / (4)

Simplify...
x^2 = 20.25

Take the square-root of each side to get just "x" on the left...
sqrt(x^2) = sqrt(20.25)

Simplify and solve...
x = 4.5
____________________

Next one...
x^2 - 2x - 24 = 0

This time you're going to factor...
So you ask yourself what are some factors of 24 that will add or subtract to equal -2?

There's... 12 and 2, but they won't add/subtract to make -2
But........... 6 and 4 make a good pair because "4 - 6 = -2"

So, your factors should look like this...
(x - 6) (x + 4) = 0
(hint: you can check to see if this is correct by multiplying out the factors and making sure you get "x^2 - 2x - 24")

Now set each factor equal to "0" and solve individully (you'll get two answers but that's okay)
x - 6 = 0
x + 4 = 0

Then solve...
x = 6
x = -4
_______________________

Last one...
For this one, I would use the "quadratic equation" which is...

x = ( -b (+/-) sqrt(b^2 - 4ac)) / 2a

with an equation that looks like...
ax^2 + bx + c = 0

So, in this case your a, b, and c are...
a = 3,
b = -1
c = - 4

Now just plug in the numbers and solve...

x = (-(-1) (+/-) sqrt( (-1)^2 - 4(3)(-4))) / (2*3)
x = (1 (+/-) sqrt(1 + 48)) / 6
x = (1 (+/-) 7) / 6

Once again you will get two answers...
x = (1 + 7) / 6 = 8/6 = 4/3
and
x = (1 - 7) / 6 = -6/6 = -1
_________________________

Hope it helped!

Answer 2

4x^2 - 81 = 0

4x^2 = 81

X^2 = 81/4 take square root of each side

x = +/- (square root 81 /4) = 9/2 = +/ - 4 1/2


x^2 - 2x -24 + 0 Factor this

(x -6)(x + 4) = 0

x -6 = 0

x = 6

x + 4 = 0

x = -4

3x^2 - x -4 = 0 Factor this

(3x -4)(x + 1)

3x -4 = 0

3x = 4

x = 4/3

x + 1 = 0

x = -1

Answer 3

4x² = 81
x² = 81/4
x = ± 9 / 2

(x - 6).(x + 4) = 0
x = 6, x = - 4

3x² - x - 4 = 0
(3x - 4).(x + 1) = 0
x = - 4 / 3, x = - 1

Answer 4

x^2-81/4=0 so (x+9/2)(x-9/2)=0 so x=-9/2 or x=9/2
x^2-2x-24=0 or (x-6)(x+4)=0 so x=6 or x=-4
3x^2-x-4=0 or (3x-4)(x+1)=0 so x=-1 or x=4/3