Question

Solve the inequality by graphing. \(x^2+6 x<-3\)

Short answer

The solution set of the inequality is \( x \) belongs to real numbers.

Step-by-step solution

Rewriting the inequality into standard form

To simplify the process of solving, we rewrite the inequality \( x^2+6x < -3 \) in standard form by bringing all terms to one side, resulting in \( x^2 + 6x + 3 > 0 \)

Finding the roots of the quadratic equation

The roots of the equation \( x^2 + 6x + 3 = 0 \) can be found by either factoring, completing the square, or using the quadratic formula. However, this expression cannot be factored easily, so we use the quadratic formula: for \( ax^2 + bx + c = 0 \), the roots are given by \( x = [-b ± sqrt(b^2 - 4ac)] / 2a \). After plugging in the coefficients \( a = 1, b = 6, and c = 3 \), we find that the roots are complex (as the discriminant \( b^2 - 4ac \) is less than zero), indicating that the parabola does not intersect the x-axis and it's either entirely above or below the x-axis.

Determining the shape and position of the parabola.

Because the coefficient of \( x^2 \) is positive, the parabola opens upwards. Also, since the equation has no real roots, it lies completely above the x-axis. As such, the inequality \( x^2 + 6x + 3 > 0 \) is true for all values of \( x \).

Checking validity

As a check, we select a random value of \( x \) and verify whether the inequality holds true. For example, we can choose \( x = 0 \). Substituting the value 0 in the inequality \( x^2 + 6x + 3 > 0 \) gives 3 which indeed is greater than 0.

Key concepts

Solving Inequalities by Graphing

When dealing with quadratic inequalities, one of the most intuitive methods of solving them is by using a graph. To do this, you need to first rewrite the inequality into a form that can be used to easily identify the shape of the curve. For example, in the inequality \(x^2 + 6x < -3\), we rewrite it as \(x^2 + 6x + 3 > 0\) by rearranging terms. This allows us to graph the parabola and determine where it lies relative to the x-axis.

By graphing, you can visually assess where the parabola is situated, whether it's above, below or on the x-axis. Since our inequality asks when the expression is greater than zero, we're looking for the parts of the graph that lie above the x-axis. This graphical representation makes complex inequalities easier to interpret and solve.

Quadratic Equations

A quadratic equation is a polynomial equation in the form \(ax^2 + bx + c = 0\). It represents a parabola when graphed on a coordinate plane. The coefficients \(a\), \(b\), and \(c\) determine the shape and position of the parabola.

• The sign of \(a\) tells us whether the parabola opens upwards \((a > 0)\) or downwards \((a < 0)\).
• The vertex of the parabola gives the minimum or maximum point, found using \(-b/(2a)\).
• The solutions, or roots, of the equation determine the x-intercepts, found using the quadratic formula when factoring isn't possible.

Understanding the parts and properties of a quadratic equation helps in graphing and solving inequalities associated with it.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is negative. The discriminant is part of the quadratic formula, expressed as \(b^2 - 4ac\). When it's less than zero, it indicates no real roots exist and the parabola doesn't intersect the x-axis. Instead, the roots are complex numbers, involving imaginary numbers.

In our example, the quadratic \(x^2 + 6x + 3 = 0\) has a negative discriminant, showing complex roots. This suggests that the entire parabola is either completely above or below the x-axis. The absence of real roots simplifies solving the inequality, as there's no need for test points between intercepts since they don't exist. This understanding is crucial when determining the regions where the inequality holds true.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Each part of the formula plays a specific role:

• \(-b/(2a)\) determines the axis of symmetry of the parabola.
• The expression \(b^2 - 4ac\) is called the discriminant, which indicates the nature of roots: real and distinct, real and repeated, or complex.
• \(\pm\) shows there are usually two solutions.

In our example, the formula reveals complex roots, highlighting the inequality \(x^2 + 6x + 3 > 0\) is true for all \(x\). Using the quadratic formula allows solving equations effectively, especially when they can't be factored easily.