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A quadratic equation is a polynomial equation of the second degree. It usually looks like this: \(ax^2 + bx + c = 0\). Solving it means finding the values of \(x\) (or any variable) that make the equation true. There are a few methods to solve quadratic equations:

• Factoring
• Completing the square
• Using the quadratic formula
• Graphing

For the equation \(7d^{2}+5=0\), the steps are different. Let's look at what was done.
The goal is to isolate and solve for the variable \(d\). To do this, you work step-by-step through a logical sequence.
First, you isolate the quadratic term by subtracting constants or dividing by coefficients. Then, if the right-hand side of the equation is negative, it suggests there might be an imaginary number involved.
Finally, you take the square root of both sides to solve for \(d\).

Imaginary numbers come into play when solving quadratic equations that have negative numbers inside the square root symbol. The imaginary unit is represented by \(i\), where \(i = \sqrt{-1}\).
Here's a quick recap on how imaginary numbers work:

• \(i^{2} = -1\)
• Multiplying by \(i\) rotates a direction in the complex plane.
• Any real number multiplied by \(i\) is imaginary.

When solving the quadratic equation \(7d^{2}=-5\), taking the square root of both sides involves an imaginary number.
You would rewrite \( \sqrt{-5}\) as \( \sqrt{5} \cdot i\). So the solution for \(d\) would be \(d = \pm \sqrt{5/7} \cdot i\). This means there are two solutions: one positive imaginary and one negative imaginary.

To solve the quadratic equation, the first thing to do is isolate the quadratic term. This means you want to get the term with \(d^2\) on one side of the equation by itself.
Here’s how you isolate the quadratic term step by step:

• Start with the equation \(7d^{2} + 5 = 0\).
• Subtract 5 from both sides: \(7d^{2} = -5\).
• Then divide both sides by 7: \(d^{2} = -5/7\).

Once isolated, you're ready to solve for \(d\). Next, take the square root of both sides, which introduces the need for imaginary numbers when the right-hand side is negative.
This method sets the foundation to find the complex solutions for the variable \(d\).