Very nice. Sometimes, I have trouble getting started, and you always help me with that. From there, Ican solve.

@SpencersAcademy

3 ай бұрын

Awesome! I am really happy to hear that.

Sweet 👍

@SpencersAcademy

2 ай бұрын

Thank you

(x-3)^4=16 Taking square root on both sides, (x-3)^2=+4 or (x--3)^2=-4 If (x-3)^2=4 taking square root again, x-3=+2 or x-3= -2 x=3+2=5 or x=3-2=1 If (x-3)^2=-4 taking square root x-3=+ or - 2i If x-3= 2i then.x=3+2i If x-3 = -2i then x=3-2i So the solutions are 5,3,3+2i and 3-2i

@SpencersAcademy

3 ай бұрын

Fantabulous. That's a good approach.

or just u=x-3 u^4=16 recall if x^n=m, then x=nthroot(m) e^{2ikpi/m} where k is an integer u=2e^{ikpi/2} x-3=+-2, +-2i x=3+-2, 3+-2i

5 and 1 is the he answer. Actually got it in seconds 16 is 4^2 which is the same as 2^4 or (-2)^4 then cancelling out 4 at the exponent at both sides x-3=2 or x-3=-2 and finally x=5 and 1

@CyberFlare-fn9kn

3 ай бұрын

You miss out in imaginary solutions with such a method

@lechaiku

3 ай бұрын

@caleboji4857 You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

Observing that the 4th root of 16 is +/-2, I found the real roots, 1 and 5, by inspection. Finding the remaining 3 roots requires a considerable amount of extra work, which the video demonstrated.

@SpencersAcademy

2 ай бұрын

That's a nice observation. I must commend you for that.

@davidbrown8763

2 ай бұрын

Many thanks. Have a nice day. NB. I neglected to state that the 4th root of 16 is +/- 2, not just 2, which I have now corrected. Hence x - 3 = 2, giving x = 5 and x - 3 = - 2, giving x= 1. That is how I was able to obtain the real roots by inspection.

1;5;3-2i;3+2i

@SpencersAcademy

3 ай бұрын

Very correct.

[(x-3)²=y] y² = 16 |y| = 4 -> y=4 or y=-4 (x-3)² = 4; (x-3)² = -4 [x-3=z] z²=4; z²=-4 z=2; z=-2; z=2i; z=-2i x=2+3; x=-2+3; x=2i + 3; x= -2i + 3 x=5; 1; 3+2i; 3-2i

@SpencersAcademy

3 ай бұрын

You nailed it, man. Bravo!!!

two solutions are obvious: x=5 and x=1

5 and 1

@SpencersAcademy

3 ай бұрын

You are right, man. But this is a fourth-degree equation. So, it was supposed to have four roots (values of x).

@lechaiku

3 ай бұрын

@harrymatabal8448 You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

(X-3)4=2to 4 X-3=2. X=5

@lechaiku

3 ай бұрын

@psycholoogdrs.hansjacobs9816 You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

(x-3)⁴ = 16 x ? ========= For real solution (x-3)⁴ = 16 (x-3)⁴ = (±2)⁴ x-3 = ±2 x - 3 = 2 --> x = 5 x - 3 = -2 --> x = 1 x = 1 & 5 7:13

(X-3)^4= 2^4 X-3 = +-2 X=3+2 Or X=3-2

@SpencersAcademy

2 ай бұрын

Nice one. X also has two other values.

X-3=+/-2

@lechaiku

3 ай бұрын

You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

6

Look at this shorter method. (x-3)^4 = 2^4 => x-3 = 2 x = 2+3 = 5

@nerd5865

3 ай бұрын

1, 3+2i, 3-2i

@lechaiku

3 ай бұрын

@ShimelesTeacher You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

(5-3)^4 = 16

The solution is too long. The shortest solution is to take the 4th root of both sides. Hence, X-3=2. X= 2+3=5; X=5. To check whether 5 is the correct value of X, substitute the X value to the original equation, (5-3)^4= 16, 2^4=16, 16=16. Therefore, 5 is the only value of X.

@lechaiku

3 ай бұрын

@gregc.mariano9226 You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

Bro it could be done in seconds

@SpencersAcademy

3 ай бұрын

I wouldn't disagree.

Way too over-complicated approach. And definitely not an Maths Olympiad question.

@SpencersAcademy

3 ай бұрын

I would appreciate it if you would show me your approach.

@DrAbdulRehman1989

2 ай бұрын

Rewrite 16 as 2^4 and raise both sides to power 1/4; you will get: x-3 = 2

Too long. 4th root of LHS=±2; Hence x=1 or 5. i is not relevant here.

@SpencersAcademy

3 ай бұрын

Remember, this is a fourth-degree equation. You don't just take the fourth root of both sides. If you do that, you're just gonna have just two values of x, instead of four.

@lechaiku

3 ай бұрын

@Onoelo23gf You always should remember that any power of x tells you how many solutions should be. In this particular case (x^4) there are 4 roots of a given equation. So we have 2 real solutions and 2 complex solutions. If you have the math problem where the domain of x is not specified, you must show all solutions. If the domain is x = R you can avoid the complex solutions.

X=5