1 Find all real solutions to the following equations a 5 t

1. Find all real solutions to the following equations:

(a). 5 – t = 2(t + 1)^2

Answers: t = _________________________________

(b). x^4 + 5x^2 – 1 = 0 (Hint: Make a substitution which would turn this into a quadratic equation)

Answers: x = _________________________________

a ) Given that

5 - t = 2 ( t +1)²

5 - t = 2 ( t² + 2t + 1)

5 - t = 2t² + 4t + 2

2t² + 5t -3 = 0

( 2t -1 ) (t + 3 ) = 0

t = 1 /2 , t = -3

Hence,

t = 1 /2 ,-3

b )

Given that

x⁴ + 5x² -1 = 0

let x² = t

then

(x²)² +5x² -1 = 0

t² + 5t -1 = 0

a = 1,b = 5,c = -1

t = ( -b ± (b² - 4ac)^1/2) /2a

then

t = ( -5 ± (25 - 4.1.-1)^1/2 ) /2.1

t = (-5 ± (29)^1/2) / 2

t = (-5 + (29)^1/2) / 2 , t = (-5 - (29)^1/2) / 2

t = 0.19 , t = -5.19

t = x²

x = t^1/2

If t = 0.19 then x = (0.19)^1/2

x = 0.435

If t = -5.19 then x = (-5.19)^1/2

x = - 2.28

Hence,

x = 0.435 ,-2.28