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 there is a sqared and a square root, I am confused on how to go about it. Could you please head me in the right direction?
I don't see a square root anywhere.
Let's start by rewriting it as
3x -1 = (x-4)^2
Then turn it into a binomial equation with zero on one side.
3x -1 = x^2 -8x + 16
x^2 -11x + 17 = 0
That does not factor easily, so use the quadratic equation to solve.
x = (1/2)[11 +/- sqrt(121 - 68)]
= (1/2)[11 +/- sqrt53]
One of the roots is x = 9.14
To solve the equation √(3x-4) = (x-4)^2 - 3, you can start by squaring both sides of the equation. This will help eliminate the square root:
(√(3x-4))^2 = ((x-4)^2 - 3)^2
Simplifying further,
(3x-4) = ((x-4)^2 - 3)^2
Next, expand the right side of the equation:
(3x-4) = (x^2 - 8x + 16 - 3)^2
(3x-4) = (x^2 - 8x + 13)^2
To simplify further, square the binomial (x^2 - 8x + 13):
(3x-4) = x^4 - 16x^3 + 106x^2 - 208x + 169
Now, rearrange the equation:
x^4 - 16x^3 + 106x^2 - 208x + 169 - (3x-4) = 0
Combine like terms:
x^4 - 16x^3 + 106x^2 - 208x + 165 = 0
Unfortunately, this equation cannot be solved by simple factoring or any other elementary algebraic techniques. It requires more advanced methods such as factoring by grouping, synthetic division, or using numerical methods.
To solve the equation √(3x-4) = (x-4)^2-3, there are two methods you can use depending on what you prefer:
Method 1: Square both sides of the equation
By squaring both sides of the equation, you can eliminate the square root sign. However, keep in mind that squaring both sides may introduce extraneous solutions, so you will need to check your solution(s) afterward.
Here's the step-by-step process:
1. Start with the equation: √(3x-4) = (x-4)^2-3
2. Square both sides of the equation to eliminate the square root: [(√(3x-4))^2] = [(x-4)^2-3]^2
Simplify: 3x - 4 = [(x-4)^2-3]^2
3. Expand and simplify the right side by squaring the binomial [(x-4)^2]:
3x - 4 = (x^2 - 8x + 16 - 3)^2
Simplify: 3x - 4 = (x^2 - 8x + 13)^2
4. Expand and simplify the right side by squaring the binomial: (x^2 - 8x + 13)^2
Simplify: 3x - 4 = x^4 - 16x^3 + 90x^2 - 208x + 169
5. Rearrange the equation into standard form (x^4 - 16x^3 + 90x^2 + (3x - 208)x + (169 - 4) = 0):
x^4 - 16x^3 + 90x^2 - 3x^2 + 3x - 208x + 169 - 4 = 0
Simplify: x^4 - 16x^3 + 87x^2 - 205x + 165 = 0
6. Now you have a quartic equation, which can be challenging to solve. You can attempt to solve it by factoring, using the rational root theorem, or numerical methods.
Method 2: Square both sides of the equation and simplify
In this method, you square both sides of the equation and simplify the resulting equation before solving for x.
Here's the step-by-step process:
1. Start with the equation: √(3x-4) = (x-4)^2-3
2. Square both sides of the equation to eliminate the square root: [(√(3x-4))^2] = [(x-4)^2-3]^2
Simplify: 3x - 4 = (x^2 - 8x + 13)^2
3. Expand and simplify the right side by squaring the binomial: (x^2 - 8x + 13)^2
Simplify: 3x - 4 = x^4 - 16x^3 + 90x^2 - 208x + 169
4. Rearrange the equation by moving all terms to one side:
x^4 - 16x^3 + 90x^2 + (-3x - 208x) + 169 - 4 - (3x - 4) = 0
Simplify: x^4 - 16x^3 + 87x^2 - 205x + 165 = 0
5. Now you have a quartic equation. You can attempt to solve it by factoring, using the rational root theorem, or numerical methods.
Remember, solving quartic equations can be challenging, and there may not always be a straightforward algebraic solution.