Mister Exam

Other calculators


sqrt(x^2-x-1)-sqrt(x)=1

sqrt(x^2-x-1)-sqrt(x)=1 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src]
   ____________            
  /  2              ___    
\/  x  - x - 1  - \/ x  = 1
$$- \sqrt{x} + \sqrt{\left(x^{2} - x\right) - 1} = 1$$
The graph
Rapid solution [src]
                                                                                      _______________________________________________________________________________________________________________
                                                                                     /            _____________                                                                                      
                                                                                    /            /       _____                                                                                       
                                                                                   /            /      \/ 489                                  8                                         16          
                                                                                  /    8 - 2*3 /   5 + -------  + ----------------------------------------------------------- - -------------------- 
                   _________________________________________________             /           \/           9                 _________________________________________________          _____________ 
                  /            _____________                                    /                                          /            _____________                                 /       _____  
                 /            /       _____                                    /                                          /            /       _____                                 /      \/ 489   
                /            /      \/ 489              16                    /                                          /            /      \/ 489              16             3*3 /   5 + -------  
               /    4 + 2*3 /   5 + -------  + --------------------          /                                          /    4 + 2*3 /   5 + -------  + --------------------      \/           9     
              /           \/           9              _____________         /                                          /           \/           9              _____________                         
             /                                       /       _____         /                                          /                                       /       _____                          
            /                                       /      \/ 489         /                                          /                                       /      \/ 489                           
           /                                   3*3 /   5 + -------       /                                          /                                   3*3 /   5 + -------                          
         \/                                      \/           9        \/                                         \/                                      \/           9                             
x1 = 1 + ----------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------
                                      2                                                                                              2                                                               
$$x_{1} = \frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + \frac{8}{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}} + 8}}{2} + 1 + \frac{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}}{2}$$
x1 = sqrt(-2*(sqrt(489)/9 + 5)^(1/3) - 16/(3*(sqrt(489)/9 + 5)^(1/3)) + 8/sqrt(16/(3*(sqrt(489)/9 + 5)^(1/3)) + 2*(sqrt(489)/9 + 5)^(1/3) + 4) + 8)/2 + 1 + sqrt(16/(3*(sqrt(489)/9 + 5)^(1/3)) + 2*(sqrt(489)/9 + 5)^(1/3) + 4)/2
Sum and product of roots [src]
sum
                                                                                 _______________________________________________________________________________________________________________
                                                                                /            _____________                                                                                      
                                                                               /            /       _____                                                                                       
                                                                              /            /      \/ 489                                  8                                         16          
                                                                             /    8 - 2*3 /   5 + -------  + ----------------------------------------------------------- - -------------------- 
              _________________________________________________             /           \/           9                 _________________________________________________          _____________ 
             /            _____________                                    /                                          /            _____________                                 /       _____  
            /            /       _____                                    /                                          /            /       _____                                 /      \/ 489   
           /            /      \/ 489              16                    /                                          /            /      \/ 489              16             3*3 /   5 + -------  
          /    4 + 2*3 /   5 + -------  + --------------------          /                                          /    4 + 2*3 /   5 + -------  + --------------------      \/           9     
         /           \/           9              _____________         /                                          /           \/           9              _____________                         
        /                                       /       _____         /                                          /                                       /       _____                          
       /                                       /      \/ 489         /                                          /                                       /      \/ 489                           
      /                                   3*3 /   5 + -------       /                                          /                                   3*3 /   5 + -------                          
    \/                                      \/           9        \/                                         \/                                      \/           9                             
1 + ----------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------
                                 2                                                                                              2                                                               
$$\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + \frac{8}{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}} + 8}}{2} + 1 + \frac{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}}{2}$$
=
                                                                                 _______________________________________________________________________________________________________________
                                                                                /            _____________                                                                                      
                                                                               /            /       _____                                                                                       
                                                                              /            /      \/ 489                                  8                                         16          
                                                                             /    8 - 2*3 /   5 + -------  + ----------------------------------------------------------- - -------------------- 
              _________________________________________________             /           \/           9                 _________________________________________________          _____________ 
             /            _____________                                    /                                          /            _____________                                 /       _____  
            /            /       _____                                    /                                          /            /       _____                                 /      \/ 489   
           /            /      \/ 489              16                    /                                          /            /      \/ 489              16             3*3 /   5 + -------  
          /    4 + 2*3 /   5 + -------  + --------------------          /                                          /    4 + 2*3 /   5 + -------  + --------------------      \/           9     
         /           \/           9              _____________         /                                          /           \/           9              _____________                         
        /                                       /       _____         /                                          /                                       /       _____                          
       /                                       /      \/ 489         /                                          /                                       /      \/ 489                           
      /                                   3*3 /   5 + -------       /                                          /                                   3*3 /   5 + -------                          
    \/                                      \/           9        \/                                         \/                                      \/           9                             
1 + ----------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------
                                 2                                                                                              2                                                               
$$\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + \frac{8}{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}} + 8}}{2} + 1 + \frac{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}}{2}$$
product
                                                                                 _______________________________________________________________________________________________________________
                                                                                /            _____________                                                                                      
                                                                               /            /       _____                                                                                       
                                                                              /            /      \/ 489                                  8                                         16          
                                                                             /    8 - 2*3 /   5 + -------  + ----------------------------------------------------------- - -------------------- 
              _________________________________________________             /           \/           9                 _________________________________________________          _____________ 
             /            _____________                                    /                                          /            _____________                                 /       _____  
            /            /       _____                                    /                                          /            /       _____                                 /      \/ 489   
           /            /      \/ 489              16                    /                                          /            /      \/ 489              16             3*3 /   5 + -------  
          /    4 + 2*3 /   5 + -------  + --------------------          /                                          /    4 + 2*3 /   5 + -------  + --------------------      \/           9     
         /           \/           9              _____________         /                                          /           \/           9              _____________                         
        /                                       /       _____         /                                          /                                       /       _____                          
       /                                       /      \/ 489         /                                          /                                       /      \/ 489                           
      /                                   3*3 /   5 + -------       /                                          /                                   3*3 /   5 + -------                          
    \/                                      \/           9        \/                                         \/                                      \/           9                             
1 + ----------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------
                                 2                                                                                              2                                                               
$$\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + \frac{8}{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}} + 8}}{2} + 1 + \frac{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}}{2}$$
=
                                                                                 _______________________________________________________________________________________________________________
                                                                                /            _____________                                                                                      
                                                                               /            /       _____                                                                                       
                                                                              /            /      \/ 489                                  8                                         16          
                                                                             /    8 - 2*3 /   5 + -------  + ----------------------------------------------------------- - -------------------- 
              _________________________________________________             /           \/           9                 _________________________________________________          _____________ 
             /            _____________                                    /                                          /            _____________                                 /       _____  
            /            /       _____                                    /                                          /            /       _____                                 /      \/ 489   
           /            /      \/ 489              16                    /                                          /            /      \/ 489              16             3*3 /   5 + -------  
          /    4 + 2*3 /   5 + -------  + --------------------          /                                          /    4 + 2*3 /   5 + -------  + --------------------      \/           9     
         /           \/           9              _____________         /                                          /           \/           9              _____________                         
        /                                       /       _____         /                                          /                                       /       _____                          
       /                                       /      \/ 489         /                                          /                                       /      \/ 489                           
      /                                   3*3 /   5 + -------       /                                          /                                   3*3 /   5 + -------                          
    \/                                      \/           9        \/                                         \/                                      \/           9                             
1 + ----------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------
                                 2                                                                                              2                                                               
$$\frac{\sqrt{- 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} - \frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + \frac{8}{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}} + 8}}{2} + 1 + \frac{\sqrt{\frac{16}{3 \sqrt[3]{\frac{\sqrt{489}}{9} + 5}} + 2 \sqrt[3]{\frac{\sqrt{489}}{9} + 5} + 4}}{2}$$
1 + sqrt(4 + 2*(5 + sqrt(489)/9)^(1/3) + 16/(3*(5 + sqrt(489)/9)^(1/3)))/2 + sqrt(8 - 2*(5 + sqrt(489)/9)^(1/3) + 8/sqrt(4 + 2*(5 + sqrt(489)/9)^(1/3) + 16/(3*(5 + sqrt(489)/9)^(1/3))) - 16/(3*(5 + sqrt(489)/9)^(1/3)))/2
Numerical answer [src]
x1 = 3.60742059854082
x1 = 3.60742059854082
The graph
sqrt(x^2-x-1)-sqrt(x)=1 equation