how do you find a quadratic equation if you are given the solutions such as x=2/3 and x=-4 right out your answer

To find a quadratic equation when given the solutions, you can start by using the fact that the solutions of a quadratic equation are the values of 'x' for which the equation is equal to zero.

In this case, you are given two solutions: x = 2/3 and x = -4.

The general form of a quadratic equation is: ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants.

To find the equation, you need to substitute each solution into the quadratic equation and solve for the constants 'a', 'b', and 'c'.

For the solution x = 2/3, you substitute it into the quadratic equation:
a(2/3)^2 + b(2/3) + c = 0

Simplifying this equation:
(4a + 2b)/9 + c = 0

Similarly, for the solution x = -4, you will substitute it into the quadratic equation:
a(-4)^2 + b(-4) + c = 0

Simplifying this equation:
16a - 4b + c = 0

Since you have two unknowns - 'a', 'b', and 'c' - you need more equations to solve for each variable.

However, you can use the fact that if 'x' is a solution, then (x - root) is a factor of the quadratic equation. Therefore, the product of these factors will give you the equation.

For the given solutions, the factors are (x - 2/3) and (x + 4), corresponding to (x - root).

To find the quadratic equation, you multiply these factors:
(x - 2/3)(x + 4) = 0

Expanding this equation:
(x^2 + 4x - (2/3)x - 8/3) = 0

Combining like terms:
x^2 + (10/3)x - 8/3 = 0

Therefore, the quadratic equation with the given solutions x = 2/3 and x = -4 is:
x^2 + (10/3)x - 8/3 = 0