How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution?

if the answer is negative it has no real # of solutions; if it is positive it has 2 real # of solutions; if its 0 it means 1

2x quared+-4=0

To determine the number of solutions of a quadratic equation, also known as a second-degree polynomial equation of the form ax^2 + bx + c = 0, you need to consider the discriminant (D), which is the value inside the square root in the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

1. If the discriminant (D = b^2 - 4ac) is positive, the equation will have two distinct real solutions. In this case, the graph of the quadratic equation will intersect the x-axis at two points.

2. If the discriminant is zero (D = 0), the equation will have exactly one real solution. The graph of the quadratic equation will touch the x-axis at one point; these are called "repeated" or "double" roots.

3. If the discriminant is negative (D < 0), the equation will have no real solutions. The graph of the quadratic equation will not intersect the x-axis. Instead, it will only have complex solutions (conjugate pairs involving the imaginary unit "i").

If you are given a solution of a quadratic equation, you can determine the equation by using backward reasoning.

For example, if the solution is x = a, then the equation can be written as (x - a)(x - a) = 0, which simplifies to x^2 - 2ax + a^2 = 0. Therefore, the quadratic equation could be x^2 - 2ax + a^2 = 0.

Regarding your last question, it is not possible to have different quadratic equations with the same solution, as the quadratic equation is unique for a given set of solutions (roots).

To determine the number of solutions a quadratic equation will have, you can look at its discriminant. The discriminant is the expression inside the square root in the quadratic formula, which is b² - 4ac, where a, b, and c are the coefficients of the quadratic equation (in the form ax² + bx + c = 0).

1. If the discriminant is greater than zero (b² - 4ac > 0), then the quadratic equation will have two distinct real solutions.
2. If the discriminant is equal to zero (b² - 4ac = 0), then the quadratic equation will have one real solution (referred to as a "double root" or "repeated root").
3. If the discriminant is less than zero (b² - 4ac < 0), then the quadratic equation will have no real solutions. In this case, the solutions will be complex conjugates.

Now, if you are given the solution(s) of a quadratic equation, you can work backward to find the equation itself. Let's break it down into two scenarios:

1. If you are given two distinct real solutions, you can use the fact that the sum and product of the roots of a quadratic equation relate to its coefficients. Let's say the solutions are x1 and x2. Then the quadratic equation can be represented as (x - x1)(x - x2) = 0. Expanding and simplifying this expression will give you the quadratic equation.

2. If you are given a repeated root (double root), let's say x1, the quadratic equation can be written as (x - x1)² = 0. Expanding and simplifying this expression will yield the quadratic equation.

Now, can different quadratic equations have the same solution? Yes, it is possible. Since the quadratic equation can be represented in general form as ax² + bx + c = 0, where a, b, and c are the coefficients, different values of a, b, and c can lead to the same solution(s). However, the specific coefficients will differ.