Solve for x in each of the following, x E C. x^4 - 7 = 6x^2The answer is +- i, +- 7. I don't know how to get the answer. I can part of it..

Expert Answers
hala718 eNotes educator| Certified Educator

x^4 - 7 = 6x^2

First let us move 6x^2 to the left side:

==> x^4 - 6x^2 -7 = 0

Now let us factor:

(x^2 - 7)(x^2 +1) = 0

==> x^2 - 7 = 0 ==> x = +-sqrt7

==> x^2 + 1 = 0 ==> x= +- i

Then the solution is:

x= { sqrt7, -sqrt7, i, -i}

giorgiana1976 | Student

We'll solve this equation using substitution technique. So, let's note x^2 by t.

If x^2 = t => x^4 = (x^2)^2 = t^2

We'll re-write the equation in t:

t^2 - 7 = 6t

We'll subtract 6t both sides:

t^2 - 6t - 7 = 0

We'll apply the quadratic formula:

t1 = [6 + sqrt(6^2 - 4*1*(-7))]/2*1

t1 = (6 + sqrt64)/2

t1 = (6+8)/2

t1 = 14/2

t1 = 7

t2 = (6-8)/2

t2 = -2/2

t2 = -1

But t = x^2

7= x^2

We'll raise both sides to the power (1/2):

7^(1/2) = (x^2)^(1/2)

+sqrt 7 = x

x = -sqrt7

t = -1

x^2 = -1

x = sqrt(-1)

But i^2 = -1, so sqrt(-1) = i

x = +i

x = -i

So, the number of complex solutions of the equation is 4 and the solutions are:

{-sqrt 7 ; +sqrt 7 ; -i ; +i}.

neela | Student

x^4-7 = 6x^2.

To solve the equation.

Subtract 6x^2 from both sides of the equatiob and we get:

x^2 -6x^2-7 = 0.

y^2-6y -7 = 0. Where y = x^2. Now we solve the quadratic equation in y.

y^2-7y +y -7 = 0

y(y-7) +1(y-7) = 0.

y -7 = 0 or y+1 = 0

(y-7)(y+1) = 0

y-7 = 0 gives y = x^2 = 7.

So x= sqrt7 or x2 = -sqrt7

y+1 = 0 gives y = x^2 = -1. So x3 = sqrt(-1) = i Or x4 = -sqrt(-1)

= -i.

So the solutons are sqrt7 , -sqrt7,  i or i