1.) Why are there usually two solutions in quadratic equations?
2.) Under what situation would one or more solutions of a rational equation be unacceptable?
If putting the found "solution" cannot be put back into the original equation without violating fundamental math rules, then it is not a solution. This usually involves haveing a root in the denominator.
There are two solutions because there are usually two values of the variable which will satisfy the conditions of the original governing equation.
•Why are there usually two solutions in quadratic equations? Are there any cases you have more than two? How about only one or none? Give examples.
1.) Quadratic equations typically have two solutions because the graph of a quadratic function is a parabola, which can intersect the x-axis at two points, one on each side of the vertex. The solutions represent the x-coordinates of these points of intersection.
However, there are cases where a quadratic equation may have more than two solutions. This can occur when the quadratic equation has a double root, meaning that the parabola touches the x-axis at a single point. In these cases, the quadratic equation has two identical solutions.
For example, the equation x^2 = 4 has two solutions: x = 2 and x = -2. The parabola y = x^2 intersects the x-axis at these two points.
2.) In rational equations, there are situations where one or more solutions may be unacceptable. This occurs when the solutions lead to a division by zero, which violates fundamental mathematical rules. When a solution makes the denominator of the rational equation equal to zero, it is considered an unacceptable solution.
For example, consider the rational equation 1/(x-2) = 3. If we solve this equation by multiplying both sides by (x-2), we get 1 = 3(x-2), which simplifies to x - 2 = 1/3. Adding 2 to both sides gives x = 7/3.
However, if we substitute x = 2 back into the original equation, we get 1/0 = 3, which is not defined. Therefore, x = 2 is an unacceptable solution for the given rational equation.
There are usually two solutions in quadratic equations because a quadratic equation represents a parabola, which typically intersects the x-axis at two points. The solutions correspond to the x-coordinates of these points of intersection.
However, there are cases where a quadratic equation may have more than two solutions. This occurs when the parabola intersects the x-axis at multiple points. For example, the equation x^2 - 4 = 0 has four solutions (-2, 2, i√2, -i√2), as it represents a parabola that intersects the x-axis at two real points and two imaginary points.
On the other hand, there are cases where a quadratic equation may have only one solution. This happens when the parabola is tangent to the x-axis at a single point. For example, the equation x^2 - 2x + 1 = 0 has one solution, x = 1, as the parabola intersects the x-axis at a single point.
Lastly, there are cases where a quadratic equation has no real solutions. This occurs when the parabola does not intersect the x-axis. For example, the equation x^2 + 1 = 0 has no real solutions, as the parabola never crosses the x-axis. However, it does have two complex solutions, i.e., x = i and x = -i.