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To solve quadratic equations like the one in this exercise, a very effective method is using the Quadratic Formula. This formula provides a straightforward way to find the solutions for equations that conform to the general quadratic form \( ax^2 + bx + c = 0 \). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use the formula, we must first identify the coefficients \(a\), \(b\), and \(c\) from the equation. For example, in our problem, we have \(a = 2\), \(b = 6\), and \(c = 7\). These numbers are crucial because they slot into the different parts of the formula, allowing you to solve for \(x\).

• The \(\pm\) symbol in the formula denotes that there are generally two solutions to consider for most quadratic equations.
• The term under the square root, \(b^2 - 4ac\), is known as the discriminant, and it determines the nature of the roots (real or complex).

Once you've filled in these coefficients and performed the arithmetic, you'll get the solutions for \(x\).

Quadratic equations can sometimes yield complex solutions, especially when the discriminant \(b^2 - 4ac\) is negative. In this scenario, the square root of a negative number is involved. Enter complex numbers!Complex numbers take the form \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). In our exercise, simplifying \(\sqrt{-20}\) leads to a complex number, as follows:

• Recognize \(\sqrt{-20}\) as \(2i\sqrt{5}\).
• This transformation is based on separating the negative sign as \(i^2\).

If you ever find a negative number under the square root in quadratic solutions, expect to involve \(i\). Once simplified, the outcome combines real and imaginary components, giving two solutions, as we saw: \(x = -1.5 \pm 0.5i\sqrt{5}\). Complex solutions expand our understanding, moving beyond real numbers to a broader number system.

Simplifying equations is an essential step and is most beneficial after using the Quadratic Formula. Let's see why and how:The Quadratic Formula may sometimes give us solutions with unnecessary complexity. By simplifying, we make solutions easier to comprehend. In our step-by-step procedure, after finding \(x = \frac{-6 \pm 2i\sqrt{5}}{4}\), further simplification was performed to yield \(x = -1.5 \pm 0.5i\sqrt{5}\).Here’s how you can achieve it:

• First, calculate the basic operations in the formula, like squaring \(b\) or multiplying products within the square root.
• Consider expressing complex numbers in simplest terms by factoring out common numbers or terms, here dividing each component by 2.

Simplification not only tidies up the final solution but also aids in understanding the structure and components of the result better. Always take this final step to refine your solutions.