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? Explain. Provide your
classmate’s with one or two solutions with which they must create a quadratic equation.
So what is your thinking? We will be happy to critique your thinking.
Usually a quadratic equation will be squared but it depends on the power of x.
b^2 - 4ac (the discrimant) determines number of solutions.
if > 0, 2 solutions
if = 0, 1 solution (a double solution)
if < 0, 0 (real) solutions (but 2 imaginary)
Given solutions x = 2, x = -4; equation is:
(x-2)(x+4) = 0 or x^2 + 2x - 8 = 0
Different equations only in the sense they are a multiple of the original one. x^2 + 3x + 5 = 0 and
2x^2 + 6x + 10 = 0 are multiples of each onther and have the same solutions.
To determine the number of solutions of a quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants, you can use the discriminant (denoted as Δ). The discriminant is calculated as Δ = b^2 - 4ac.
1. If Δ > 0, the quadratic equation will have two distinct real solutions. This happens when the discriminant is positive, indicating that the equation's graph intersects the x-axis at two points.
2. If Δ = 0, the quadratic equation will have one real solution. The discriminant being zero suggests that the equation's graph touches the x-axis at a single point. This solution is called a repeated or double root.
3. If Δ < 0, the quadratic equation will have no real solutions. In this case, the discriminant is negative, implying that the equation's graph doesn't intersect the x-axis at any point. However, it will have two complex (non-real) solutions.
Now, if you are given the solutions x1 and x2, and you need to find the quadratic equation, you can rewrite the equation in factored form using the given solutions. Consider the two solutions (roots) as x1 and x2:
(x - x1)(x - x2) = 0
Expanding this equation will yield the quadratic equation with these solutions.
Yes, it is possible to have different quadratic equations with the same solution(s). This can occur when the quadratic equations have the same roots but different coefficients. For example, the equations 2x^2 + 3x + 1 = 0 and 4x^2 + 6x + 2 = 0 both have the solution x = -0.5. They are different equations but share the same solution.
Now, let me provide you with a solution, and you can create a quadratic equation using it.
Solution: x = 3
Using this solution, you can create a quadratic equation as follows:
Step 1: Subtract 3 from both sides: x - 3 = 0
Step 2: Square the equation: (x - 3)^2 = 0
Step 3: Expand the equation: x^2 - 6x + 9 = 0
Therefore, the quadratic equation with the given solution x = 3 is x^2 - 6x + 9 = 0.