Write down the equation whose roots are:
i) two more than
ii) the square of
iii) three times as much as the roots of the equation 4x^2-x+2=0
To find the equation whose roots satisfy the given conditions, let's break it down step by step:
i) Two more than the roots:
Let's assume the roots are denoted by r1 and r2. If the roots are two more than r1 and r2, respectively, the equations can be written as:
x = r1 + 2
x = r2 + 2
Now, to find the equation, we need to eliminate x. We can do this by substituting the values of r1 and r2. Therefore, the equations become:
(x - 2) = r1
(x - 2) = r2
The equation with these roots can be written as:
(x - 2)(x - 2) = 0
Expanding this equation, we get:
x^2 - 2x - 2x + 4 = 0
Simplifying further gives us:
x^2 - 4x + 4 = 0
ii) The square of the roots:
If the roots are the square of r1 and r2, the equations can be written as:
x^2 = r1^2
x^2 = r2^2
Now, again, let's eliminate x. We can do this by substituting the values of r1 and r2. Therefore, the equations become:
x^2 - r1^2 = 0
x^2 - r2^2 = 0
The equation with these roots can be written as:
(x - r1)(x + r1) = 0
Expanding this equation using the difference of squares, we get:
x^2 + r1x - r1x - r1^2 = 0
Simplifying further results in:
x^2 - r1^2 = 0
iii) Three times as much as the roots of the equation:
If the roots of the equation 4x^2 - x + 2 = 0 are multiplied by 3, the equations can be written as:
x = 3r1
x = 3r2
Now, let's eliminate x. We can do this by substituting the values of r1 and r2. Therefore, the equations become:
x/3 = r1
x/3 = r2
The equation with these roots can be written as:
(x/3 - r1)(x/3 - r2) = 0
Expanding this equation, we get:
(x^2 - 3r1x + 3r1x - 9r1^2) / 9 = 0
Simplifying further results in:
x^2 - 9r1^2 = 0
In summary, the equations with the given conditions are as follows:
i) Two more than the roots:
x^2 - 4x + 4 = 0
ii) The square of the roots:
x^2 - r1^2 = 0
iii) Three times as much as the roots of the equation 4x^2 - x + 2 = 0:
x^2 - 9r1^2 = 0