Question

( x + 5 )^4 - 10 (x - 3)^2 + 9 =0

Answer 1

4x+20-10x-60+9=0

-6x-49=0

-6x=49

x=-8.1

Answer 2

To solve the equation (x + 5)^4 - 10(x - 3)^2 + 9 = 0, we can use a substitution to simplify the equation. Let's set y = (x + 5)^2. Substituting this into the equation, we have:

y^2 - 10(x - 3)^2 + 9 = 0

Expanding (x - 3)^2, we get:

y^2 - 10(x^2 - 6x + 9) + 9 = 0

Simplifying further, we have:

y^2 - 10x^2 + 60x - 90 + 9 = 0 y^2 - 10x^2 + 60x - 81 = 0

Now, we have a quadratic equation in terms of y. Let's solve it:

y^2 - 10x^2 + 60x - 81 = 0

We can use the quadratic formula to find the values of y:

y = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 60, and c = -10x^2 - 81. Plugging in these values, we get:

y = (-60 ± √(60^2 - 4(1)(-10x^2 - 81))) / (2(1))

Simplifying further:

y = (-60 ± √(3600 + 40x^2 + 324)) / 2 y = (-60 ± √(40x^2 + 3924)) / 2 y = (-30 ± √(10x^2 + 981))

Now, we can substitute y back into the equation y = (x + 5)^2:

(x + 5)^2 = (-30 ± √(10x^2 + 981))

Taking the square root:

x + 5 = ±√(-30 ± √(10x^2 + 981))

Solving for x:

Case 1: x + 5 = √(-30 + √(10x^2 + 981)) x = -5 + √(-30 + √(10x^2 + 981))

Case 2: x + 5 = -√(-30 + √(10x^2 + 981)) x = -5 - √(-30 + √(10x^2 + 981))

These are the solutions to the equation (x + 5)^4 - 10(x - 3)^2 + 9 = 0. Please note that it might not be possible to find an exact solution to this equation algebraically, and numerical methods such as approximation or graphing may be necessary to find the actual values of x.