Question

In Problems \(7-22,\) solve each inequality. 18\. \(x^{2}+2 x+4>0\)

Short answer

The solution is all real numbers.

Step-by-step solution

- Understand the inequality

The inequality given is a quadratic inequality: \(x^{2} + 2x + 4 > 0\)

- Determine the nature of the quadratic equation

To analyze the inequality, first consider the quadratic equation \(x^{2} + 2x + 4 = 0\). Calculate its discriminant, \(D = b^2 - 4ac\), where \(a = 1\), \(b = 2\), and \(c = 4\).

- Calculate the discriminant

\(D = 2^2 - 4*1*4\) \(D = 4 - 16\) \(D = -12\)

- Analyze the discriminant

The discriminant \(D = -12\) is less than zero, which means the quadratic equation \(x^{2} + 2x + 4 = 0\) has no real roots. Hence, the parabola opens upwards and does not intersect the x-axis.

- Conclusion from the parabolic graph

Since the parabola opens upwards and lies entirely above the x-axis, the quadratic expression \(x^{2} + 2x + 4\) is always greater than zero for all real values of \(x\).

- State the solution

The solution to the inequality \(x^{2} + 2x + 4 > 0\) is all real numbers.

Key concepts

quadratic equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\). Quadratic equations are important in algebra and appear in various real-world scenarios.
The general form ensures that the highest exponent of the variable \(x\) is 2. For example, in the equation \(x^2 + 2x + 4 = 0\), \(a=1\), \(b=2\), and \(c=4\).
To solve quadratic equations, we often use methods like factoring, completing the square, or the quadratic formula given by \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\].
Quadratic equations can have 0, 1, or 2 real roots depending on the value of the discriminant, which we'll delve into next.

discriminant

The discriminant is a special part of the quadratic formula, represented by \(D = b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation.
The value of the discriminant tells us:

• If \(D > 0\), the quadratic equation has two distinct real roots.
• If \(D = 0\), the quadratic equation has exactly one real root (a repeated root).
• If \(D < 0\), the quadratic equation has no real roots; instead, it has two complex roots.

For the inequality \(x^2 + 2x + 4 > 0\), we calculated the discriminant as \(D = 2^2 - 4*1*4 = -12\), which is less than zero.
This means there are no real roots for the equation \(x^2 + 2x + 4 = 0\).We will now connect this to the concept of parabolic graphs.

real numbers analysis

Real numbers cover all numbers on the number line, including rational and irrational numbers. When analyzing solutions for inequalities involving quadratic expressions, we often investigate if the solution applies to all real numbers.
Given our inequality \(x^2 + 2x + 4 > 0\), we determined that the discriminant is less than zero, indicating no real roots.
We then analyzed the graph of the quadratic equation. Since it opens upwards and intersects the x-axis at no points, the quadratic expression \(x^2 + 2x + 4\) is always above the x-axis for all real values of \(x\). Hence, the solution to the inequality is all real numbers.

parabolic graph

A parabolic graph represents a quadratic function and is a U-shaped curve called a parabola. Its general equation form is \(y = ax^2 + bx + c\).
The direction of the parabola (opening upwards or downwards) is determined by the coefficient \(a\). If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
For the quadratic inequality \(x^2 + 2x + 4 > 0\), the graph opens upwards because \(a = 1 > 0\).
Since the discriminant is negative, there are no x-intercepts. The entire graph lies above the x-axis.
Therefore, the inequality holds true for all real values of \(x\),which confirms that the expression \(x^2 + 2x + 4\)is always positive.