Write a quadratic equation in the variable x having the given numbers as solutions. Type the equation in standard form, ax^2+bx+c=0
-sqrt3, 3sqrt3
sum of roots = 2√3
product of roots = -9
equation:
x^2 - 2√3 x - 9 = 0
To write a quadratic equation with -√3 and 3√3 as solutions, we can use the fact that the solutions of a quadratic equation are the values of x that make the equation equal to zero.
Given that the solutions are -√3 and 3√3, we know that when x = -√3, the equation is equal to zero, and when x = 3√3, the equation is also equal to zero.
To get the equation, we use the fact that if a number is a solution to a quadratic equation, then its conjugate (opposite sign for square root) is also a solution. In this case, if -√3 is a solution, then √3 is also a solution, and if 3√3 is a solution, then -3√3 is also a solution.
This means that the factors of the quadratic equation are (x - (-√3)) and (x - 3√3). Simplifying, we get (x + √3) and (x - 3√3).
Now, multiplying these factors together will give us the quadratic equation.
(x + √3)(x - 3√3) = 0
Expanding this equation, we get:
x^2 - 3√3x + √3x - 3(√3)^2 = 0
Simplifying further, we have:
x^2 - 3√3x + √3x - 9 = 0
Combining like terms, we get:
x^2 - 2√3x - 9 = 0
So, the quadratic equation in standard form with -√3 and 3√3 as solutions is:
x^2 - 2√3x - 9 = 0.