SOLUTION: Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. 1. {{{x^4 + (x^2)*sqrt(2) - 2 = 0}}} 2. {{{pi(1 + r)^2

Algebra ->  Radicals -> SOLUTION: Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places. 1. {{{x^4 + (x^2)*sqrt(2) - 2 = 0}}} 2. {{{pi(1 + r)^2       Log On


   

Question 1207633: Find the real solutions of each equation. Use a calculator to express any solutions rounded to two decimal places.

1. x%5E4+%2B+%28x%5E2%29%2Asqrt%282%29+-+2+=+0

2. pi%281+%2B+r%29%5E2+=+2+%2B+pi%281+%2B+r%29
Answer by semsee45(1) About Me  (Show Source):
You can put this solution on YOUR website!

QUESTION 1


Let +u=x%5E2+, so +x%5E4+%2B+x%5E2+%2A+sqrt%282%29+-+2+=+0+ becomes +u%5E2+%2B+sqrt%282%29+u+-+2+=+0+.

Solve for u using the quadratic formula, where +a+=+1+, +b+=+sqrt%282%29+ and +c+=+-2+:



+u+=+%28-sqrt%282%29+%2B-+sqrt%2810%29%29%2F2+

Substitute back in +u=x%5E2+:

+x%5E2+=+%28-sqrt%282%29+%2B-+sqrt%2810%29%29%2F2+

As we cannot take the square root of a negative number, the only valid real solution is:

+x%5E2+=+%28-sqrt%282%29+%2B+sqrt%2810%29%29%2F2+

Solving for x gives:

+x+=+sqrt%28%28-sqrt%282%29+%2B+sqrt%2810%29%29%2F2%29+=+highlight%280.93%29+

+x+=+-sqrt%28%28-sqrt%282%29+%2B+sqrt%2810%29%29%2F2%29+=+highlight%28-0.93%29+


QUESTION 2


Rearrange +pi%281+%2B+r%29%5E2+=+2+%2B+pi%281+%2B+r%29+ so that it is in the form +ax%5E2+%2B+bx+%2B+c+=+0+:

+pi%281+%2B+r%29%5E2+-+pi%281+%2B+r%29+-+2+=+0+

Expand:

+pi%2A%281+%2B+2r+%2B+r%5E2%29+-+pi+-+pi+%2A+r+-+2+=+0+
+pi+%2B+2+%2A+pi+%2A+r+%2B+pi+%2A+r%5E2+-+pi+-+pi+%2A+r+-+2+=+0+
+pi+%2A+r%5E2+%2B+pi+%2A+r+-+2+=+0

Solve for r using the quadratic formula, where +a+=+pi+, +b+=+pi+ and +c+=+-2+:

+r+=+%28-pi+%2B-+sqrt%28+%28pi%29%5E2-4%2Api%2A%28-2%29+%29%29%2F%282%2Api%29+

+r+=+%28-pi+%2B-+sqrt%28+%28pi%29%5E2%2B8%2Api%29%29%2F%282%2Api%29+

Solve for r:

+r+=+%28-pi+%2B+sqrt%28+%28pi%29%5E2%2B8%2Api%29%29%2F%282%2Api%29+=+highlight%280.44%29+

+r+=+%28-pi+-+sqrt%28+%28pi%29%5E2%2B8%2Api%29%29%2F%282%2Api%29+=+highlight%28-1.44%29+