Find all real values of x which satisfy x2+1+x2+1=90.

Question

Abdullah Avery · Accepted Answer

Let me point out couple of substitutions that simplifies this equation:
1.

T = \sqrt{x^{2} + 1}

, then we have a quadratic equation

T^{2} + T - 90 = 0

which has two solution

T = 9 and T = - 10

. Then you continue from here (with the condition that

T \geq 1

).
2.

x = \tan θ

, then

x^{2} + 1 = \sec^{2} θ

and we get a trigonometric equation

\sec^{2} θ + \sec θ - 90 = 0 .

Similarly

x = \cot θ

is also a possible trigonometric substitution for this equation.
3.

x = sinh θ

has the property that

x^{2} + 1 = {cosh}^{2} θ

and hence we get

{cosh}^{2} θ + cosh θ - 90 = 0 .

In last two substitutions both sec and cos h are even functions, therefore we don't need to consider the negative square root as a separate case.

Alisha Chambers · Accepted Answer

Begin by puttingy=x2+1with y+y2=90or y2+y−90=0y=−1±192and y&amp;gt;0.You should find that x=±80.

Find all real values of x which satisfy