(not sure how to type square root of) square root of 3x - 4 = (x-4)^2-3
Normally, I either take the square root of both sides or I square both sides. Since one side of the equation is square root of 3x-4 and the other side is (x-4) squared - 3, I am confused on how to go about it. Could you please head me in the right direction?
square both sides.
The right side will be onerous, gather terms, combine with the right side, and you have a fourth degree polynomial that might factor.
Another way: graph y= (x-4)^2-3-sqrt(3x-4) and look for zeros. You should be able to do that on a graphical calc quickly.
Indeed, it is a more challenging problem.
If you keep the square root, chances are that you have to go to a numerical solution, such as Newton's method, or the bisection method to locate the roots. If you square both sides to eliminate the square root radical, and resulting in a quartic, you will need to eliminate false roots due to the squaring by back substituting the roots of the quartic.
In a case like this, it is much easier to make a sketch of both functions.
f1(x)=sqrt(3*x-4)-(x-4)^2+3
see
http://i263.photobucket.com/albums/ii157/mathmate/ashley1.png
and that of function 2 after squaring:
f2(x)=x^4-16*x^3+90*x^2+(-211)*x+173
see
http://i263.photobucket.com/albums/ii157/mathmate/ashley2.png
We can see two false roots between 3 and 5, while the roots at 1.8 and 6.6 appear in both graphs.
Now you can proceed to refine the roots at the proper locations by whichever appropriate method you choose.
To solve the equation (√(3x - 4) = (x - 4)^2 - 3), we need to simplify and then solve for x. Here's how you can approach it:
Step 1: Square both sides of the equation to eliminate the square root:
(√(3x - 4))^2 = ((x - 4)^2 - 3)^2
Simplifying this equation gives us:
3x - 4 = (x - 4)^4 - 6(x - 4)^2 + 3
Step 2: Expand and simplify the right side of the equation:
Expand (x - 4)^4 using the binomial expansion formula, and distribute the -6:
3x - 4 = x^4 - 4x^3 + 6x^2 - 4x^2 + 16x + 3 - 6x^2 + 24x - 24 + 3
Combine like terms:
3x - 4 = x^4 - 4x^3 - 4x^2 + 40x - 22
Step 3: Rearrange the equation to get it in the form of a polynomial equation set equal to zero:
x^4 - 4x^3 - 4x^2 + 37x - 18 = 0
Now, you can try solving this polynomial equation using various methods like factoring, synthetic division, or using numerical methods like graphing or a calculator.
I hope this helps you understand the process of solving the equation (√(3x - 4) = (x - 4)^2 - 3). Let me know if you have any further questions!