Problem 91 Suppose $$ 2 x^{2}+3 x+c>... [FREE SOLUTION]

Chapter 2: Problem 91

Suppose $$ 2 x^{2}+3 x+c>0 $$ for every real number $x$. Show that $c>\frac{9}{8}$

Short Answer

Expert verified

In order for the inequality $2x^2 + 3x + c > 0$ to hold true for every real number x, we analyze the discriminant as $螖 = 3^2 - 4(2)(c) = 9 - 8c$. The quadratic function will have no real roots if the discriminant is less than 0. Hence, we have $9 - 8c < 0$, which, when solved for c, gives us $c > \frac{9}{8}$.

Step by step solution

Identify the quadratic function

The quadratic function here is $f(x) = 2x^2 + 3x + c$. This function opens upwards (since the coefficient of $x^2$ is positive). If it has no real roots, then it will have no points of intersection with the x-axis. That will ensure that the inequality is always greater than 0.

Analyze the discriminant

In order to determine if a quadratic function has any real roots, we need to compute the discriminant. The discriminant of a quadratic of the form $ax^2+bx+c$ is given by $螖 = b^2 - 4ac$. For our problem, we have: 螖 = $3^2 - 4(2)(c)$ 螖 = $9 - 8c$ To have no real roots, the discriminant must be less than 0. Therefore, we need: 9 - 8c < 0

Solve for c

Now, let's solve the inequality to find the range for c: 9 - 8c < 0 -8c < -9 (||------------------) Divide both sides by -8 and reverse the sign: c > $\frac{9}{8}$ So, in order for the inequality $2x^2 + 3x + c > 0$ to hold for every real number x, the value of c must be greater than $\frac{9}{8}$.

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Key Concepts

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

Quadratic Function

A quadratic function is a type of polynomial that takes the general form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a eq 0$. This ensures the graph of the function is a parabola. The coefficient $a$ determines the direction the parabola opens:

If $a > 0$, the parabola opens upwards, creating a U-shape.
If $a < 0$, the parabola opens downwards, making an upside-down U-shape.

In this discussion, we have the quadratic function $f(x) = 2x^2 + 3x + c$. It opens upwards because the coefficient of $x^2$ is positive ($2 > 0$). This particular parabola does not touch the x-axis because it has no real roots when $c > \frac{9}{8}$, meaning the function is always positive.

Discriminant

The discriminant is a key component in determining the nature of the roots of a quadratic equation. For a given quadratic function $ax^2 + bx + c$, the discriminant $\Delta$ is calculated using the formula:\[\Delta = b^2 - 4ac\]The value of the discriminant helps us understand whether the quadratic has:

Two distinct real roots if $\Delta > 0$.
One real root if $\Delta = 0$, meaning the graph touches the x-axis at one point.
No real roots if $\Delta < 0$, indicating the graph does not cross the x-axis.

For our function $f(x) = 2x^2 + 3x + c$, the discriminant is calculated as $\Delta = 9 - 8c$. For the parabola to not intersect the x-axis, ensuring the inequality $2x^2 + 3x + c > 0$ holds, we require $\Delta < 0$. When solved, this leads to the condition $c > \frac{9}{8}$.

Real Roots

Real roots of a quadratic function are the x-values where the function crosses or touches the x-axis. These occur when the quadratic equation $ax^2 + bx + c = 0$. The discriminant plays a central role in realizing the number and nature of these real roots:

Two real roots exist if the discriminant is positive, as the parabola intersects the x-axis at two distinct points.
A single real root exists if the discriminant equals zero; the parabola touches the x-axis at exactly one point.
No real roots are present if the discriminant is negative, meaning the parabola sits entirely above or below the x-axis.

In the problem associated with the equation $2x^2 + 3x + c > 0$ for all $x$, achieving no real roots is crucial. This requirement ensures the function remains consistently positive, which is satisfied when $c > \frac{9}{8}$. This prevents the parabola from touching or crossing the x-axis.

Inequalities

Inequalities describe the range of values that satisfy certain conditions, often involving parts of algebraic expressions. When dealing with inequalities like $2x^2 + 3x + c > 0$, it is important to understand they define sets of values for $x$ and other parameters (like $c$).

An inequality $f(x) > 0$ implies that the function's output is greater than 0 for all values in its domain.
Similarly, $f(x) < 0$ suggests function outputs are less than zero through its domain.

In solving the given quadratic inequality $2x^2 + 3x + c > 0$, the solution $c > \frac{9}{8}$ indicates the conditions under which this inequality holds for every real number $x$. By ensuring $c$ is greater than $\frac{9}{8}$, the quadratic curve remains above the x-axis, thus ensuring the inequality is always satisfied.

91影视

Suppose $$ 2 x^{2}+3 x+c>0 $$ for every real number \(x\). Show that \(c>\frac{9}{8}\)

Short Answer

Step by step solution

Identify the quadratic function

Analyze the discriminant

Solve for c

Key Concepts

Quadratic Function

Discriminant

Real Roots

Inequalities

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