91Ó°ÊÓ

In the context of a quadratic equation, a coefficient is a numerical or constant value placed in front of variables like x. When you look at a quadratic equation in the form of\(ax^2 + bx + c = 0\), the letters a, b, and c are coefficients.

Each coefficient has a specific role:

• a: The leading coefficient multiplies \(x^2\).
• b: This coefficient multiplies x.
• c: The constant term, which stands alone.

To solve a quadratic equation using the quadratic formula, you need to correctly identify these coefficients. For example, given the function \(f(x) = -2x^2 + 3x - 4\), the coefficients are:

• a = -2
• b = 3
• c = -4

Knowing the coefficients allows you to plug these values into the quadratic formula.

Complex solutions arise when the discriminant in the quadratic formula is negative. The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\).

If this value is negative, the equation has complex (or imaginary) solutions because the square root of a negative number is not a real number. Instead, it involves the imaginary unit \(i\), where \(i = \sqrt{-1}\).

For example, in our solved exercise:

• The discriminant is calculated as \(3^2 - 4(-2)(-4) = 9 - 32 = -23\).
• Since -23 is negative, the solutions will be complex.
• We write this as \(\sqrt{-23} = i\sqrt{23}\).

The final solutions for x are:

• \(\frac{3}{4} - \frac{i\sqrt{23}}{4}\)
• \(\frac{3}{4} + \frac{i\sqrt{23}}{4}\)

Complex solutions indicate that the graph of the quadratic function does not intersect the x-axis.

A quadratic equation is a second-order polynomial equation in a single variable x, with the general form \(ax^2 + bx + c = 0\). Quadratic equations are fundamental in algebra and are widely used to model various real-world phenomena.

The solutions or 'roots' of a quadratic equation can be found using several methods, one of which is the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula provides the solutions by taking into account the coefficients a, b, and c.
Here is the process step-by-step using our example \(f(x) = -2x^2 + 3x - 4\):

• Identify the coefficients: \(a = -2\), \(b = 3\), \(c = -4\).
• Plug these into the quadratic formula: \( x = \frac{-(3) \pm \sqrt{(3)^2 - 4(-2)(-4)}}{2(-2)} \).
• Simplify under the square root: \( x = \frac{-3 \pm \sqrt{9 - 32}}{-4} = \frac{-3 \pm \sqrt{-23}}{-4} \).

This leads to complex solutions since the discriminant is negative. Understanding quadratic equations and their solutions helps in graphing and interpreting many algebraic expressions.