Question

In Problems \(7-22,\) solve each inequality. \(19.4 x^{2}+9<6 x\)

Short answer

The inequality has no real solutions.

Step-by-step solution

- Move all terms to one side

Begin by moving all terms to one side of the inequality to set it to zero. Subtract \(6x\) from both sides: \[19.4 x^{2} + 9 - 6x < 0.\]

- Reorder the terms

Reorder the terms in the standard quadratic form \(ax^2 + bx + c < 0\): \[19.4 x^{2} - 6x + 9 < 0.\]

- Find the roots of the equation

To find the critical points, solve the corresponding equation for equality: \[19.4 x^{2} - 6x + 9 = 0.\] Use the quadratic formula: \[x = \frac{{-b \, \text{±} \, \sqrt{{b^2 - 4ac}}}}{{2a}},\] where \(a = 19.4, b = -6, c = 9\).

- Calculate the discriminant

Compute the discriminant \(\text{{Δ}}\): \[Δ = b^2 - 4ac = (-6)^2 - 4(19.4)(9) = 36 - 696 = -660.\] Since the discriminant is negative, there are no real roots.

- Analyze the quadratic expression

Since there are no real roots and the coefficient \(a = 19.4\) is positive, the quadratic expression \(19.4 x^{2} - 6x + 9\) is always positive for all real values of \x\. Thus, the inequality \[19.4 x^{2} - 6x + 9 < 0\] has no real solutions.

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This formula can be expressed as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}.\] To use it, you need the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. Simply plug them into the formula to find the solutions or roots of the equation.

• The term \(b^2 - 4ac\) under the square root is called the discriminant.
• The \(\pm \) symbol means you will get two results: one with addition and one with subtraction.

Knowing the roots of the quadratic equation helps us understand where the function intersects the x-axis. This is crucial for solving inequalities like the one in this problem.

Discriminant

The discriminant (denoted as \(\Delta\)) is an important part of the quadratic formula. It's found inside the square root: \[\Delta = b^2 - 4ac.\] The value of the discriminant tells us about the nature of the roots:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (the roots are repeated).
• If \(\Delta < 0\), there are no real roots (the roots are complex numbers).

In our exercise, the discriminant was calculated to be \(-660\). Since this value is negative, it indicates that the quadratic equation has no real roots. Instead, it has two complex roots.

No Real Roots

When the discriminant is negative, the quadratic equation has no real roots. This means the graph of the quadratic function does not intersect the x-axis. For our inequality \19.4x^2 - 6x + 9 < 0\, since \(\Delta = -660 \) (a negative value), we concluded that there are no real roots.
Additionally:

• Because the coefficient of \(x^2\) (which is \(a = 19.4\)) is positive, the parabola opens upwards.
• This tells us that the quadratic expression \19.4x^2 - 6x + 9\ is always positive for all real values of \(x\).

Therefore, the inequality \(19.4x^2 - 6x + 9 < 0\) has no real solutions because the expression can never be less than zero.
Understanding these concepts will help you solve similar problems with confidence.