1. Explain in detail how to determine the value of the independent variable in a quadratic relation if the value of the dependent variable is known.
Please check my answer -
You would substitute the value for the dependent variable, then you would solve for the independent variable.
(is it right?)
2. What is the greatest number of solutions a quandratic equation can have? Explain, with an example, why all of the solutions to the equations may not be reasonable answers to the original problem.
My answer-
A quandratic equations can have either no solution, one solution or two solutions.
Can someone please help me with the answer to this question too? Please explain it also.. i find it so hard. >.<
Thanks!
1. The independent variable is the "x" in the quadratic equation ax^2 + bx + c = y (the dependent variable)
If you know y, then solve
ax^2 + bx + (c-y) = 0
c-y is a new constant, c'
The equation can be solved by completing the square or using the equation
x = [-b +/- sqrt(b^2-4ac')]/2a
2. You are right; the greatest number of solutions is two. There can be one solution in some cases. It is possible that all solutions contain imaginary numbers (the square root of a negative number), in which case there are no REAL solutions.
It is possible that only one of the two solutions is a reasonable solution to the problem. For example, if you throw a ball and ask when it hits the ground,
y = at - bt^2,
there will be two solutions; one will be the time t=0 when you threw it.
Variable of 3/23=j/3
1. To determine the value of the independent variable in a quadratic relation when the value of the dependent variable is known, you would follow these steps:
Step 1: Start with the standard form of a quadratic equation: y = ax^2 + bx + c, where y represents the dependent variable, x represents the independent variable, and a, b, and c are constants.
Step 2: Substitute the known value of the dependent variable into the equation. For example, if you know that y = 25, you would replace y with 25 in the equation: 25 = ax^2 + bx + c.
Step 3: Rearrange the equation to solve for the independent variable, x. In this case, you would have: ax^2 + bx + c = 25.
Step 4: Depending on the specific question, you might need to use additional information or algebraic techniques to solve for x. This could involve factoring, completing the square, or using the quadratic formula.
2. The greatest number of solutions a quadratic equation can have is two. However, it is important to consider the practical context of the problem to determine if all the solutions are reasonable answers.
Here's an example to illustrate this:
Let's say we have the quadratic equation: x^2 - 9 = 0. By factoring, we can rewrite it as (x - 3)(x + 3) = 0. So, the solutions are x = 3 and x = -3.
Now, let's consider a problem where x represents the length of a side of a square. It wouldn't make sense to have a negative length, so the solution x = -3 would not be a reasonable answer in this case.
Similarly, in some real-life scenarios, a quadratic equation may have no solution. For instance, if you have the equation x^2 + 9 = 0, there are no real values for x that satisfy this equation. In the context of a real-world problem, this would mean there is no solution to the given situation.
Therefore, while a quadratic equation can have different numbers of solutions, not all solutions are always reasonable or applicable to the original problem. It's important to carefully consider the context to determine which solutions are valid in practical terms.