How do you know if a quadratic equation will have one, two, or no solutions?Please give examples. How do you find a quadratic equation if you are only given the solution? Please give an example.Is it possible to have different quadratic equations with the same solution? Explain and please give example. Provide your classmate’s with one or two solutions with which they must create a quadratic equation.
You can tell by the discriminant
a x^2 + b x + c = 0
look at
b^2 - 4 ac
If positive, then two real solutions
If zero, then vertex on x axis and both solutions are the same
If negative, then no real solutions, solutions are complex numbers
for the rest, work up from factors
for example
(x-1)(x-2) = 0 two real x = 1 or 2
x^2 -3 x + 2 = 0
for example
(x-2)(x-2)
x^2 - 4 x + 4 = 0 one solution (or two both the same)
(x-i)(x+i) = 0
x^2 - i^2 = x^2+1 = 0 no real solutions
To determine the number of solutions of a quadratic equation, we can look at its discriminant, which is the value inside the square root of the quadratic formula. The discriminant can be used to classify the solutions as follows:
1. If the discriminant is positive (greater than zero), the quadratic equation will have two distinct real solutions. For example, consider the equation x^2 - 4x + 4 = 0. The discriminant is (b^2 - 4ac) = (-4)^2 - 4(1)(4) = 0, which is positive. Hence, this equation has two real solutions: x = 2.
2. If the discriminant is zero, the quadratic equation will have one real solution (a double root). For instance, consider the equation x^2 - 4x + 4 = 0. The discriminant is once again (b^2 - 4ac) = (-4)^2 - 4(1)(4) = 0. Since the discriminant is zero, this equation has one real solution with a multiplicity of two: x = 2.
3. If the discriminant is negative, the quadratic equation will have no real solutions. Instead, it will have two complex solutions. For example, consider the equation x^2 + 4x + 5 = 0. The discriminant is (b^2 - 4ac) = (4)^2 - 4(1)(5) = -4. Since the discriminant is negative, this equation has no real solutions. Instead, its solutions are complex numbers.
To find a quadratic equation given its solutions, we can use the factored form of a quadratic equation. If the solutions are x = p and x = q, where p and q are real numbers, the quadratic equation can be expressed as (x - p)(x - q) = 0. For example, if the solutions are 3 and -2, the quadratic equation can be written as (x - 3)(x + 2) = 0.
Yes, it is possible to have different quadratic equations with the same solutions. This is because the quadratic equation represents a curve, and different equations can produce the same curve. For example, consider the equations x^2 - 5x + 6 = 0 and 2x^2 - 10x + 12 = 0. Both equations have the solutions x = 2 and x = 3, but they are different quadratic equations.
Now, let's provide you and your classmates with two solutions to create a quadratic equation:
Solutions: -4 and 1
To create a quadratic equation with these solutions, we can use the factored form. Since the solutions are -4 and 1, the quadratic equation can be written as (x + 4)(x - 1) = 0. Expanding this equation gives us x^2 + 3x - 4 = 0, which is the quadratic equation with -4 and 1 as solutions.