91Ó°ÊÓ

Understanding quadratic equations is an important step in solving many algebraic problems. A quadratic equation is a polynomial equation of the second degree, generally written in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The equation can take various forms, but its defining feature is the \(x^2\) term.

The graph of a quadratic equation is a parabola. It opens upwards if \(a > 0\) and downwards if \(a < 0\). Solving a quadratic equation means finding the values of \(x\) that satisfy the equation, also known as the roots of the equation.

• Quadratic equations can have zero, one, or two real roots.
• The nature of the roots depends on the discriminant \(b^2 - 4ac\).

Tip: Always check if the quadratic can be factored before moving to more complex solutions like the quadratic formula.

Domain constraints are conditions that are put on the variable \(x\) within a function. They determine the set of permissible inputs for which the function is defined. Inverting a function, like finding \(g^{-1}(5)\), often requires evaluating these constraints.

In the given exercise, the function is defined as \(g(x) = x^2 + 4x\) with the restriction \(x \geq -2\). This means the function only considers values of \(x\) that are greater than or equal to \(-2\). Therefore, any solution to our equation must respect this constraint.

• Domain constraints limit the valid solutions when finding inverse functions.
• Always verify solutions against domain conditions to ensure they are acceptable.

Checking the domain is crucial. In this exercise, one potential solution \(x = -5\) is invalid because it does not meet the constraint \(x \geq -2\). Only \(x = 1\), which satisfies this domain, is valid.

The quadratic formula is a reliable method for solving any quadratic equation \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the possible values of \(x\) by substituting the coefficients \(a\), \(b\), and \(c\) from the equation.

The term \(b^2 - 4ac\) under the square root is called the discriminant. It determines the nature of the roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is one real root (or a repeated root).
• If negative, there are no real roots, only complex roots.

In our example, by using \(a = 1\), \(b = 4\), and \(c = -5\), we found that the discriminant is 36, which is positive. This indicates two possible real roots. After calculating, we checked which root falls within the domain constraint of \(x \geq -2\) to find the valid solution \(x = 1\).