Irrational equations "with an unknown" are those which include a square root where the radicand includes the unknown factor.

Example 1:  . This equation can be expressed graphically by making it equal to 0 and drawing the graph of the equation: y = LHS of the equation.

Look at the following window.

1.-Note that the curve cuts the X-axis when x = 2. Write the value of x in the box at the bottom or change the value using the arrows. Check that x = 2 is the solution of the equation. 

2.-Review and write down in your exercise book how this type of equation is solved numerically.

In this case you just need to square both sides of the equation you obtain:

x + 2 = x², a quadratic equation which can be expressed as x² - x - 2 = 0 giving the solutions: x = 2 and x = -1. We saw the solution x = 2 earlier but why haven't we seen the solution x = -1? Is it really another solution of the equation?

As we have seen, when an irrational equation is solved numerically it may have solutions which cannot be expressed graphically.

With our earlier equation: we saw that x = -1 is a numerical solution but not a graphical one. But if we check this solution with the equation:
we get . This is a possibility, but usually we would only consider a positive square root as being valid, which is also the case with the program used in the window above.

 Solve the equation

If we solve it numerically, as we did in the example above, we get the simple solution:

x = -34/10 = -3.4

In the following window we can see the graph of the equation which cuts the X-axis at this value of x. The equation has both a numerical and graphical solution.

It can be explained in the same way as the example above.

1.- Solve the equation.

In this case the graphical solution is shown in the window by writing in the relevant equation. 

"Take care with the notation used. Remember that a square root is the same as to the power 1/2". "Don't forget any brackets"

If you have written the equation correctly you will have seen that the only solution is x = 2.

2.-Check that this is actually the solution which satisfies the equation. 

The equation has two solutions: x = 10 and x = 2. The solution x = 2 was obtained graphically, but the solution x = 10, was obtained in a similar way to example 1. It may not be accepted as valid as if we take the positive roots we get: 7 = 4 - 3

Change the equation given in the following window to solve the following equations graphically: 

a)

b)

c)

Check your results by finding the numerical solutions and checking their validity.