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The standard form of a quadratic equation is one of the foundational concepts in algebra. It sets the stage for solving quadratic equations accurately. The standard form is given by the equation: - \( ax^2 + bx + c = 0 \) Here:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

The equation is considered standard when all the terms are on one side of the equation, set to zero on the other side. This form is crucial because it allows for the application of the quadratic formula or factoring to find solutions.
In the problem provided, rearranging: \( 2 = -9x^2 - x \) involves moving terms, resulting in \( -9x^2 - x - 2 = 0 \). This equation is now in standard form, with coefficients \( a = -9 \), \( b = -1 \), and \( c = -2 \). This step is vital as it aligns the problem for straightforward solution finding using subsequent methods.

The discriminant is a key component of the quadratic formula method, tucked within the square root: \[ b^2 - 4ac \]. Its value tells us a lot about the roots of the quadratic equation:

• If the discriminant is positive, there are two real and distinct solutions.
• If it is zero, there is exactly one real solution (or a repeated root).
• If negative, as in this example, there are no real solutions. Instead, the solutions are complex.

In this scenario, the calculated discriminant \((-1)^2 - 4(-9)(-2) = -71\) is less than zero, indicating that the quadratic equation has complex solutions. Complex numbers arise whenever the square root of a negative number needs to be computed. This understanding is integral for determining the nature of the solutions and guides further solution steps, especially when employing the quadratic formula.

Complex solutions appear when the discriminant of a quadratic equation is negative, which means there are no real number solutions. Complex numbers can be expressed in the form \( a + bi \), where \( i \) is the imaginary unit defined by \( i^2 = -1 \).
For the quadratic equation from the exercise, using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \], results in complex solutions due to the negative discriminant \(-71\). The solution can be simplified to: \[ x = \frac{1 \pm i \sqrt{71}}{-18} \].
This is interpreted as two complex numbers:

• \( x = -\frac{1}{18} + \frac{i \sqrt{71}}{18} \)
• \( x = -\frac{1}{18} - \frac{i \sqrt{71}}{18} \)

These solutions include a real part \( -\frac{1}{18} \) and an imaginary part \( \frac{i \sqrt{71}}{18} \). Understanding how to interpret these solutions is powerful in math, providing answers to problems involving complex numbers, as no real solutions exist for this equation.