/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 85 Suppose $$ a t^{2}+5 t+4>... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 85

Suppose $$ a t^{2}+5 t+4>0 $$ for every real number \(t\). Show that \(a>\frac{25}{16}\).

Short Answer

Expert verified
For the quadratic inequality \(at^2 + 5t + 4 > 0\) to hold true for every real number t, its discriminant, \(D = 25 - 16a\), must be less than zero. Solving the inequality \(25 - 16a < 0\) gives us \(a > \frac{25}{16}\).

Step by step solution

01

Find the discriminant

Recall that the discriminant of a quadratic function is given by D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic function \(ax^2 + bx + c\). In our case, we have: \(a = a\) \(b = 5\) \(c = 4\) Plugging these values into the discriminant formula, we get: \(D = 5^2 - 4 \cdot a \cdot 4 = 25 - 16a\)
02

Apply constraints on the discriminant

Since the quadratic inequality \(at^2 + 5t + 4 > 0\) holds true for all real numbers t, it means the quadratic function must always be positive. This can only happen if the quadratic function has no real roots, which means that the discriminant must be less than zero: \(D < 0\) Substitute the expression for D: \(25 - 16a < 0\)
03

Solve the inequality for a

Now, let's rearrange the inequality to isolate the variable a: \(25 - 16a < 0\) \(-16a < -25\) \(a > \frac{25}{16}\) Thus, we have shown that for the given inequality to be true for every real number t, the coefficient a must be greater than \(\frac{25}{16}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose $$ p(x)=x^{5}+2 x^{3}+1 $$ (a) Find two distinct points on the graph of \(p\). (b) Explain why \(p\) is an increasing function. (c) Find two distinct points on the graph of \(p^{-1}\).

Verify that \((x+y)^{3}=x^{3}+3 x^{2} y+3 x y^{2}+y^{3}\).

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} t(x) $$

Explain why the composition of two rational functions is a rational function.

Find a polynomial \(p\) of degree 3 such that \(-1,2,\) and 3 are zeros of \(p\) and \(p(0)=1\).
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.