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Solving for a variable means finding the value of one variable in an equation relative to the other variables. It’s like trying to solve a mystery by finding out what one unknown value equals. In equations such as the quadratic ones, solving for a variable involves getting it by itself on one side of the equation.
Here, we were tasked with solving for the variable \( t \) in the equation \( h = \frac{1}{2} g t^{2} + v_{0} t \). This involves using algebraic techniques like rearranging and substituting values to get \( t \) alone.
Key strategies often include:

• Identifying the term with the variable you're solving for.
• Rearranging the equation, possibly involving other variables if necessary.
• Using mathematical formulas tailored to the structure of the equation.
• Simplifying calculations to obtain the desired expression for the variable.

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the second degree. A typical quadratic equation can be written as \( at^2 + bt + c = 0\). The quadratic formula \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] allows us to find the values of \( t \) that satisfy this equation.
To use the formula effectively, identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation. In our specific exercise, \( a = \frac{1}{2}g \), \( b = v_{0} \), and \( c = -h \). Substitute these values into the quadratic formula to solve for \( t \). This technique is incredibly helpful for determining the roots of any quadratic equation.

• Remember to check the discriminant \( b^2 - 4ac \) before applying the formula, as it informs the number and type of solutions.
• The solutions from the formula can help visualize possible values for real-world contexts, like finding time in motion problems.

The discriminant is a component of the quadratic formula and is key to understanding the nature of the solutions of a quadratic equation. The discriminant, represented by \( b^2 - 4ac \), determines:

• The number of real roots of the equation.
• The nature of these roots, whether they are real or complex.

Here's how to interpret the discriminant:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there’s exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the roots are complex, not real.

In our example: \[ b^2 - 4ac = v_{0}^{2} + 2gh \] By evaluating this expression, we determined the possible outcomes for \( t \), given the contextual limitations (like time being non-negative). Understanding the discriminant gives insight into the feasibility of scenarios described by the equation.

Rearranging equations is a fundamental mathematical skill that involves altering the structure of an equation to make it easier to solve for a specific variable. It's like rearranging pieces of a puzzle to see the picture you need.
In our exercise, the original equation \( h = \frac{1}{2} g t^{2} + v_{0} t \) was rearranged into the standard quadratic form \( \frac{1}{2}gt^2 + v_{0}t - h = 0 \). This step was crucial because it set up the equation for the application of the quadratic formula.
Here are some general tips when rearranging:

• Perform the same operation on both sides to maintain equality.
• Keep track of positive and negative signs, especially when moving terms across the equals sign.
• Factor and simplify terms where possible to reduce complexity.
• Double-check each step to ensure that the rearrangement is mathematically sound.

Mastering rearrangement helps in solving complex equations by breaking them down into manageable forms.