Question

\(x^2+7 x+12<0\)

Short answer

The solution to the inequality \(x^2+7x+12<0\) is \(x \in (-4, -3)\)

Step-by-step solution

Solve the corresponding equation

We first need to solve the quadratic equation \(x^2+7x+12=0\). It can be factored into \((x+3)(x+4)=0\). Setting each factor equal to zero gives roots at \(x=-3\) and \(x=-4\).

Determine the intervals based on the roots

Now that we have the roots of the quadratic equation, next step is to divide the number line into intervals based on these roots. Our intervals are \((-\infty, -4), (-4, -3), (-3, +\infty)\).

Determine the sign of quadratic expression in each interval

The sign of quadratic expression in each interval can be tested by substituting a number from each interval into the quadratic expression. For the interval \((-\infty, -4)\), let's choose \(x=-5\), which gives \(((-5)+3)((-5)+4)<0\). This is positive. For the interval \((-4, -3)\), let's choose \(x=-3.5\), which gives \(((-3.5)+3)((-3.5)+4)<0\). This is negative. And for \((-3, +\infty)\), let's choose \(x=0\), which gives \((0+3)(0+4)<0\). This is again positive.

Write down the solution

The quadratic inequality \(x^2+7x+12<0\), holds when the expression is negative. From Step 3, we see that this occurs in the interval \((-4, -3)\). So, the solution to the inequality is \(x \in (-4, -3)\).

Key concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving quadratic equations is essential for finding the points where the expression equals zero, known as the roots.

There are several methods to solve these equations:

• Factoring: This involves expressing the quadratic in terms of two binomials. For example, \(x^2 + 7x + 12 = (x+3)(x+4)\). Set each factor to zero: \(x+3=0\) and \(x+4=0\), providing roots \(x=-3\) and \(x=-4\).
• Quadratic Formula: If factoring isn't feasible, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: This involves reorganizing the quadratic expression into a perfect square trinomial.

The chosen method depends on the specific coefficients and the desired outcome, but factoring tends to be the most straightforward when manageable.

Factoring Quadratics

Factoring quadratics is a key step in solving quadratic inequalities and equations. It involves expressing the quadratic in a form that is easier to analyze.

For example, with the quadratic expression \(x^2 + 7x + 12\), we aim to rewrite it as a product of two linear factors:

• First, identify two numbers that multiply to the constant term \(12\) (the \(c\) in the standard form \(ax^2 + bx + c\)) and add up to the linear coefficient \(7\) (the \(b\)). These numbers are \(3\) and \(4\).
• The quadratic then factors neatly into \((x + 3)(x + 4)\).

Factoring is significant because it allows us to see the roots directly, which are where each factor equals zero. In our example, setting \((x + 3) = 0\) and \((x + 4) = 0\) yield roots \(x = -3\) and \(x = -4\), respectively. This strategy simplifies the solution process significantly.

Sign Analysis

Sign analysis is a crucial step in solving quadratic inequalities, where you determine how an expression behaves across different intervals of a number line. Here's how it works:

1. Start by finding the roots of the corresponding quadratic equation, which you should have from your factoring step.
2. These roots divide the number line into distinct intervals.
3. Select a test point from each interval and substitute it back into the quadratic expression.

• For feedback on positivity or negativity, compute the expression at the point.
• In our case: for \((-\infty, -4)\), test with \(x = -5\); resulting in a positive sign.
• For \((-4, -3)\), test with \(x = -3.5\); here the expression is negative.
• For \((-3, \infty)\), use \(x = 0\); again showing a positive sign.

Sign analysis helps identify where the quadratic condition \( (x^2 + 7x + 12 < 0) \) is satisfied. The inequality is true when the expression is negative, indicating the solution is within \((-4, -3)\).

Interval Notation

Interval notation is used to express the solution set of inequalities clearly and concisely. It's particularly useful in denoting the range for which a quadratic inequality holds.

Typically, interval notation uses brackets to specify closed intervals and parentheses to denote open intervals:

• Open Interval: \((a, b)\) signifies that \(a\) and \(b\) are not included in the interval. Used when the inequality is strict, such as \(<\) or \(>\).
• Closed Interval: \([a, b]\) indicates inclusion of the endpoints. This would apply for \(\leq\) or \(\geq\) inequalities.
• Half-Open Interval: Combines both, e.g., \([a, b)\) includes \(a\) but not \(b\).

For the quadratic inequality \(x^2 + 7x + 12 < 0\), the solution was found in the interval \((-4, -3)\). This means the expression is negative only between \(x = -4\) and \(x = -3\), excluding the endpoints because the inequality is strict.