can show me how to do this with out just squaring
3x^(1/2) - x - 6 = 0
how do i do this and solve for x with out just squareing and using quadratic formula. I was never taught the proof of the quadratic formula and would like to know how to do with without squareing and if squaring is the only option which i doubt to solve for x can someone show me the proof using these number step by step
thanks
you could try graphing.
change to 3√x = x+6
then split it into
y = 3√x and y = x+6
a few points on the graph would be (0,0), (1,3), (4,6) and (9,9)
the second is of course a straight line with slope of 1 and y-intercept of 6
it is immediately obvious that the two graphs cannot intersect, so your equation has no real solution.
The only other option is to square both sides of 3√x = x+6 ,
to obtain the quadratic equation x^2 + 3x + 36 = 0
which only has complex roots as shown by our graph above.
I am not sure if this helps.
If we rearrange and square each side we end up with
8x^2-12x-36=0
and by inspection we see that one solution is x=3
to find the other divide
8x^2-12x-36 by x-3
which gives
8x-12=0 as the other solution
but check my maths!!
if you want a "proof" and development of the quadratic equation, it is obtained by completing the square on
Ax + By + C = 0 as shown in
http://mathworld.wolfram.com/QuadraticEquation.html
(BTW, I notice that DrRuss has a small error in his solution.
when you square 3√x you get 9x not 9x^2)
OOOPS That what comes of solving things quickly!!
To solve the equation 3x^(1/2) - x - 6 = 0 without squaring or using the quadratic formula, we can use a technique called "factoring". First, let's rewrite the equation in a different form:
3x^(1/2) - x = 6
Next, we can rearrange the terms:
3x^(1/2) - x - 6 = 0
Now, let's try to factor this equation. Notice that the term x^(1/2) has a square root, which means it can be written as (x^(1/2))^2 = x. Substituting this into the equation gives us:
3(x^(1/2))^2 - x - 6 = 0
Simplifying further:
3x - x - 6 = 0
Next, combine like terms:
2x - 6 = 0
Now, we have an equation that only involves the variable x. Solving this equation, we can isolate x:
2x = 6
x = 6/2
x = 3
Therefore, the solution to the equation 3x^(1/2) - x - 6 = 0 is x = 3.
To summarize the steps without factoring, we initially rewrote the equation to separate the term with the square root. Then, by squaring both sides, we eliminated the square root and simplified the equation to a quadratic form. Lastly, we either factored the quadratic equation or used the quadratic formula to find the solutions. However, in this scenario, we successfully solved for x without resorting to these more advanced techniques by recognizing an opportunity for factoring.