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Chapter 4: Problem 19

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

Short Answer

Expert verified
The inequality has no real solutions.

Step by step solution

01

- 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: \[19.4 x^{2} + 9 - 6x < 0.\]
02

- Reorder the terms

Reorder the terms in the standard quadratic form \(ax^2 + bx + c < 0\): \[19.4 x^{2} - 6x + 9 < 0.\]
03

- 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: \[x = \frac{{-b \, \text{卤} \, \sqrt{{b^2 - 4ac}}}}{{2a}},\] where \(a = 19.4, b = -6, c = 9\).
04

- Calculate the discriminant

Compute the discriminant \(\text{{螖}}\): \[螖 = b^2 - 4ac = (-6)^2 - 4(19.4)(9) = 36 - 696 = -660.\] Since the discriminant is negative, there are no real roots.
05

- 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Solve the inequality \(27-x \geq 5 x+3 .\) Write the solution in both set notation and interval notation.

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-4 x^{2}-6 x+2\)

Find the function whose graph is the graph of \(y=\sqrt{x}\) but reflected about the \(y\) -axis.

Determine algebraically whether \(f(x)=\frac{-x}{x^{2}+9}\) is even, odd, or neither.

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?
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