Question

(x-5)^6+(x2-6x+7)^3=0

Answer 1

${(x - 5)}^{6} + {( {x}^{2} - 6x + 7) }^{3} = 0 \\ {(x - 5)}^{6} = - {( {x}^{2} - 6x + 7) }^{3} \\ {(x - 5)}^{2} = -{x}^{2} +6x - 7 \\ {x}^{2} - 10x + 25 = -{x}^{2} +6x -7 \\ 2{x}^{2}-16x+32=0\\{x}^{2}-8x+16=0\\{(x-4)}^{2}=0\\x=4$

Answer 2

To solve the equation $(x-5)^6 + (x^2-6x+7)^3 = 0$, we can follow these steps:

Step 1: Expand both terms Step 2: Combine like terms Step 3: Solve for x

Let's start with Step 1:

Step 1: Expand both terms

$(x-5)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 5x + 1$

$(x^2-6x+7)^3 = (x^2-6x+7)(x^2-6x+7)(x^2-6x+7)$

Now we need to multiply these terms out. It can be quite tedious, but let's do it:

$(x^2-6x+7)(x^2-6x+7) = x^4 - 12x^3 + 49x^2 - 84x + 49$

Now, we multiply this result by $x^2-6x+7$ again:

$(x^4 - 12x^3 + 49x^2 - 84x + 49)(x^2-6x+7)$

$= x^6 - 18x^5 + 121x^4 - 378x^3 + 490x^2 - 294x + 49$

Step 2: Combine like terms

Now we add the two expanded terms together:

$x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 5x + 1 + x^6 - 18x^5 + 121x^4 - 378x^3 + 490x^2 - 294x + 49 = 0$

Step 3: Solve for x

Now, we have a polynomial equation in $x$ of degree 6. Unfortunately, finding analytical solutions for such equations can be very challenging and often requires the use of numerical methods. However, we can still attempt to solve it using numerical approaches like graphing or numerical solvers.

If you have access to graphing software, you can plot the graph of the equation $y = (x-5)^6 + (x^2-6x+7)^3$ and find the x-intercepts, which represent the solutions.

Alternatively, you can use numerical solvers like Newton-Raphson, bisection method, or other root-finding algorithms to approximate the solutions. There are also software tools and programming libraries that can help you with this task.

Keep in mind that equations of high degree may have multiple solutions, and some solutions might be complex numbers. Therefore, the numerical approach will give you approximations, and you may need to check the validity of the solutions obtained.