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In algebra, isolating the variable means rearranging an equation so that the variable you're solving for is on one side of the equation by itself. This often requires performing operations like addition, subtraction, multiplication, or division on both sides of the equation. For example, when solving the equation \( d = k t^{2} \) for \( t \), we begin by isolating the \( t^{2} \) term.

To isolate the term with the variable, divide both sides of the equation by \( k \), resulting in: \[ \frac{d}{k} = t^2 \]

This step makes the term with the variable \( t^{2} \) stand alone, a common initial move in solving equations.

A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants. In this case, the equation \( d = k t^{2} \) is a form of quadratic equation where the variable is squared.

Unlike linear equations, where the highest power of the variable is 1, quadratic equations always involve the variable being squared (raised to the power of 2). Solving quadratic equations can involve factoring, completing the square, or using the quadratic formula. In our exercise specifically, we simplify solving the quadratic equations by isolating \( t^{2} \) first, transforming it into: \[ t^2 = \frac{d}{k} \]

Remember that the general structure involves a squared term making the solutions potentially more complex.

Taking the square root of both sides is a critical step in solving quadratic equations. This operation reverses squaring. Once we have the equation in the form of \( t^2 = \frac{d}{k} \), the next step is to find the value of \( t \).

To do this, take the square root of both sides: \[ t = \frac{d}{k} \]

The square root gives you the principal (positive) root. Keep in mind that square roots can have both a positive and a negative solution because \( (\frac{d}{k})^2 \) results in \( t^{2} \) regardless of whether \( t \) is positive or negative. Therefore, the proper solution is: \[ t = \frac{d}{k}, -\frac{d}{k} \]

Always consider both the positive and negative roots when solving quadratic equations unless restricted by context or specific instructions.