91Ó°ÊÓ

Understanding the reciprocal of a number is essential when solving equations that involve sums like the one in this exercise. The reciprocal of a number is quite simply the inverse of that number. Specifically, if you have a number denoted by \(x\), its reciprocal is given by \(\frac{1}{x}\). This means that if you multiply a number and its reciprocal, the result is always 1. Let's see some examples to deepen the understanding:

• If \(x = 4\), then the reciprocal is \(\frac{1}{4}\).
• For \(x = \frac{1}{2}\), the reciprocal is 2 since \(\frac{1}{\frac{1}{2}} = 2\).
• Likewise, if \(x = -3\), then the reciprocal is \(-\frac{1}{3}\).

Recognizing the reciprocal plays a crucial role in many mathematical problems, particularly when dealing with these kinds of fractional sums.
The key takeaway is that the relationship is always mutual— the number and its reciprocal when multiplied together always yield 1.

The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation typically takes the form \(ax^2 + bx + c = 0\). The quadratic formula then provides a straightforward way to find the solutions for any quadratic equation by using the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this formula:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• \(\pm\) indicates that there may be two solutions: one with addition and one with subtraction.

Using this formula allows you to determine the roots of the quadratic equation without needing any factoring.
In our exercise, the quadratic equation is \(2x^2 - 3x\sqrt{2} + 2 = 0\), where:

• \(a = 2\)
• \(b = -3\sqrt{2}\)
• \(c = 2\)

Substituting these values into the quadratic formula gives us the solutions for \(x\). Remember, the quadratic formula will always find the roots provided \(a\), \(b\), and \(c\) are known.

The discriminant is a portion of the quadratic formula that gives insight into the nature of the roots of the quadratic equation. It is given by the expression \(b^2 - 4ac\). This value indicates how many and what kind of solutions the quadratic equation has as follows:

• If the discriminant is positive, the quadratic equation has two real and distinct roots.
• If the discriminant is zero, there is exactly one real root, sometimes called a repeated or double root.
• If the discriminant is negative, the quadratic equation has no real roots; instead, it has two complex roots.

In our particular problem, calculating the discriminant involves the following steps:First, compute \((-3\sqrt{2})^2 - 4 \times 2 \times 2\):

• \((-3\sqrt{2})^2 = 18\)
• \(4 \times 2 \times 2 = 16\)
• Thus, the discriminant is \(18 - 16 = 2\)

This positive discriminant tells us there are two real and distinct roots to the equation, which aligns with the solutions given in the problem.