91Ó°ÊÓ

The quadratic formula is a crucial tool in solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the values of \( x \) that satisfy the equation by plugging in the coefficients \( a \), \( b \), and \( c \) into the formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

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

• If it is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, there are no real roots, only complex ones.

In our problem, for the equation \( x^2 + 4x - 3 = 0 \), we have \( a = 1 \), \( b = 4 \), and \( c = -3 \). The discriminant is \( 16 + 12 = 28 \), which is positive, indicating two distinct real roots. Using the formula, we find the roots to be \( x = -2 + \sqrt{7} \) and \( x = -2 - \sqrt{7} \).

Inequalities are similar to equations but instead of equality, they involve relations like less than (<) or greater than (>). Solving inequalities means finding all values of \( x \) that make the inequality true. A graphical approach often helps, as quadratic equations describe parabolas.
In our exercise, we deal with the inequality \( x^2 + 4x - 3 < 0 \) and \( x^2 + 4x - 3 > 0 \). The roots \( x = -2 + \sqrt{7} \) and \( x = -2 - \sqrt{7} \) divide the x-axis into different regions:

• Between the roots, the parabola is below the x-axis, which means the inequality \( < 0 \) is satisfied.
• Outside the roots, the parabola is above the x-axis, so the inequality \( > 0 \) holds true.

For \( x^2 + 4x - 3 < 0 \), the solution is the interval between the roots, while for \( x^2 + 4x - 3 > 0 \), the solution lies outside these roots, towards the negative and positive infinities.

Interval notation is a compact way to describe a set of numbers between two endpoints. It is particularly useful when expressing the solution to inequalities. Let's break it down:

• \((a, b)\) denotes all numbers greater than \(a\) and less than \(b\), not including \(a\) and \(b\).
• \([a, b]\) includes \(a\) and \(b\) as part of the solution set.
• Parentheses and brackets can be mixed if one endpoint is included and the other is not.

In our problem:

• The interval \((-2 - \sqrt{7}, -2 + \sqrt{7})\) is used to describe the values of \(x\) for which the inequality \(x^2 + 4x - 3 < 0\) is true.
• The union of intervals \((-\infty, -2 - \sqrt{7}) \cup (-2 + \sqrt{7}, \infty)\) represents the solution to \(x^2 + 4x - 3 > 0\).

Interval notation makes it easier to visualize and communicate the set of solutions.

In mathematics, the roots of an equation are the values of the variable that satisfy the equation when set to zero. For a quadratic equation like \( ax^2 + bx + c = 0 \), the roots are the solutions found by the quadratic formula or factoring methods.
The roots are critical; they tell us where the graph of the quadratic function, a parabola, intersects the x-axis. In our example, the roots \( x = -2 + \sqrt{7} \) and \( x = -2 - \sqrt{7} \) are the x-values where the equation equals zero.

• Understanding these roots helps solve inequalities by determining where the parabola changes from above the x-axis to below or vice versa.
• These points are not only solutions to the equation but also boundaries for the solution set of inequalities associated with the quadratic function.

By knowing the roots, you can accurately figure out where the quadratic function is positive or negative, crucial for inequalities.