91Ó°ÊÓ

When solving a quadratic inequality graphically, we can gain a visual understanding of the solutions. Start by rewriting the inequality as an equation. This helps us identify the roots of the quadratic. The roots, or x-values where the quadratic equals zero, are the points where the graph intersects the x-axis.
In this example, the equation is: \[ x^{2} + 4x + 3 = 0 \]
By factoring, we get the roots: \[ x = -3 \] and \[ x = -1 \].
Next, sketch the graph of the quadratic function. Quadratic functions generally make a U-shaped curve called a parabola. Since the coefficient of the \(x^2\) term is positive, the parabola opens upwards. Plot the roots on the x-axis and shade the region where the inequality \( x^{2} + 4x + 3 \leq 0 \) holds true.
To find where to shade, look where the parabola is below or equal to the x-axis. Between the roots, the parabola dips below the x-axis, indicating values that satisfy the inequality. Hence, the region between \( x = -3 \) and \( x = -1 \) is shaded.

Interval notation is a concise way of writing subsets of real numbers. When expressing the solution of inequalities, it tells us the range of values that satisfy the inequality.
For our quadratic inequality \( x^{2} + 4x + 3 \leq 0 \)
we found the roots \( x = -3 \) and \( x = -1 \).
We look at the intervals around these roots to see where the inequality holds:

• \( -\infty \) < x < \(-3\)
• \( -3 \leq x \leq -1 \) (where we test within this range)
• x > \(-1\)

After testing points within each interval, we find the interval \( -3 \leq x \leq -1 \) satisfies \( x^{2} + 4x + 3 \leq 0 \).
In interval notation, this solution is written as:
\[ [-3, -1] \]
This means all x-values from -3 to -1, inclusive, satisfy the inequality.

Factoring is a key algebraic method used to solve quadratic equations. It involves rewriting the quadratic in the form of \( (x - p)(x - q) = 0 \), where p and q are the roots or solutions of the equation.
Consider our equation: \( x^{2} + 4x + 3 = 0 \).
We look for two numbers that multiply to 3 (constant term) and add to 4 (coefficient of the x-term). These numbers are 3 and 1. So, we rewrite the equation as:
\[ (x + 3)(x + 1) = 0 \]
Setting each factor equal to zero gives us the solutions:

• \( x + 3 = 0 \), hence \( x = -3 \)
• \( x + 1 = 0 \), hence \( x = -1 \)

These roots tell us where the graph of the quadratic will intersect the x-axis. Factoring quadratics is an efficient way to solve equations and is particularly useful for graphing and solving inequalities.