91Ó°ÊÓ

Understanding how to solve quadratic equations is a fundamental skill in algebra. A quadratic equation is typically written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The goal is to find the values of \(x\) that make the equation true.

• The most common method is the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This formula is derived from the process of completing the square and works for any quadratic equation.
• You can also solve quadratic equations by factoring, where you express the quadratic as a product of its factors. However, this method only works easily when the equation factors neatly.
• Another way to solve is by graphing, which reveals the solutions as the points where the parabola intersects the x-axis, also known as the roots or zeros of the equation.

To solve the equation \(x^2 + 6x + 5 = 0\) using the quadratic formula:

• Identify \(a = 1\), \(b = 6\), and \(c = 5\).
• Substitute these values into the formula: \(x = \frac{-6 \pm \sqrt{6^2-4(1)(5)}}{2(1)}\).
• Simplify under the square root: \(6^2 - 4(1)(5) = 36 - 20 = 16\).
• The roots are \(x = \frac{-6 + 4}{2} = -1\) and \(x = \frac{-6 - 4}{2} = -5\).

Solving quadratic equations by the quadratic formula is a reliable method, and knowing how to use it will greatly help you in algebra.

Graphing parabolas is another essential skill, especially when dealing with quadratic equations and inequalities. A parabola is the graph of a quadratic function, \(y = ax^2 + bx + c\). Understanding its key features can help you in solving various mathematical problems.

• The vertex is the highest or lowest point of the parabola, depending on whether it opens upwards (\(a > 0\)) or downwards (\(a < 0\)).
• The x-intercepts (or the roots) are the points where the parabola crosses the x-axis. These correspond to the solutions of the equation \(ax^2 + bx + c = 0\).
• The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.

For the equation \(x^2 + 6x + 5\), the parabola opens upwards because the coefficient of \(x^2\) (\(a\)) is positive.
The roots, found previously as \(x = -1\) and \(x = -5\), indicate where the parabola crosses the x-axis.
By sketching the parabola, you can visualize these points and determine the regions where the quadratic inequality \(x^2 + 6x + 5 > 0\) holds true.

• The parabola is above the x-axis to the left of \(x = -5\).
• It also is above the x-axis to the right of \(x = -1\).

This graphical representation helps you to understand the solution set of the inequality.

Interval notation is a way of writing subsets of the real numbers in a concise, clear format. It is particularly useful when describing the solution sets of inequalities.

• Around infinite endpoints, we use round brackets, like \((-\infty, a)\), to indicate that infinity is not a number you can reach.
• Closed intervals, indicating that endpoints are included, use square brackets like \([a, b]\).
• Open intervals, where endpoints are not included, use round brackets, such as \((a, b)\).

For the quadratic inequality \(x^2 + 6x + 5 > 0\), we found that the parabola is above the x-axis outside the interval \([-5, -1]\). Therefore, the solution set is where \(x\) is less than \(-5\) or greater than \(-1\).
This translates to interval notation as: \((-\infty, -5) \cup (-1, \infty)\).
The union symbol (\(\cup\)) here means the combination of the two intervals.
Interval notation is an efficient way to express solutions for inequalities, setting the stage for more advanced mathematics.