Which Values Of $c$ Will Cause The Quadratic Equation $-x^2 + 3x + C = 0$ To Have No Real Number Solutions? Check All That Apply.A. $-5$ B. $-\frac{9}{2}$ C. $-\frac{1}{4}$ D. $1$ E.

Introduction

In mathematics, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on the quadratic equation $-x^2 + 3x + c = 0$ and determine which values of $c$ will cause the equation to have no real number solutions.

Understanding Quadratic Equations

To understand which values of $c$ will cause the quadratic equation to have no real number solutions, we need to recall the properties of quadratic equations. A quadratic equation has real number solutions if its discriminant is non-negative. The discriminant of a quadratic equation is given by the formula $b^2 - 4ac$. If the discriminant is negative, the equation has no real number solutions.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this formula, $a$, $b$, and $c$ are the constants of the quadratic equation, and $x$ is the variable. The $\pm$ symbol indicates that there are two possible solutions to the equation.

Determining the Values of c

To determine which values of $c$ will cause the quadratic equation to have no real number solutions, we need to set the discriminant of the equation to be negative. The discriminant of the equation $-x^2 + 3x + c = 0$ is given by $3^2 - 4(-1)c = 9 + 4c$. To have no real number solutions, the discriminant must be negative, so we set $9 + 4c < 0$.

Solving the Inequality

To solve the inequality $9 + 4c < 0$, we need to isolate the variable $c$. We can do this by subtracting 9 from both sides of the inequality, which gives us $4c < -9$. Then, we can divide both sides of the inequality by 4, which gives us $c < -\frac{9}{4}$.

Checking the Answer Choices

Now that we have determined the values of $c$ that will cause the quadratic equation to have no real number solutions, we can check the answer choices to see which ones satisfy the inequality $c < -\frac{9}{4}$.

A. $-5$

We can plug in $c = -5$ into the inequality $c < -\frac{9}{4}$ to see if it satisfies the inequality. Since $-5 < -\frac{9}{4}$, the answer choice $-5$ satisfies the inequality.

B. $-\frac{9}{2}$

We can plug in $c = -\frac{9}{2}$ into the inequality $c < -\frac{9}{4}$ to see if it satisfies the inequality. Since $-\frac{9}{2} < -\frac{9}{4}$, the answer choice $-\frac{9}{2}$ satisfies the inequality.

C. $-\frac{1}{4}$

We can plug in $c = -\frac{1}{4}$ into the inequality $c < -\frac{9}{4}$ to see if it satisfies the inequality. Since $-\frac{1}{4} > -\frac{9}{4}$, the answer choice $-\frac{1}{4}$ does not satisfy the inequality.

D. $1$

We can plug in $c = 1$ into the inequality $c < -\frac{9}{4}$ to see if it satisfies the inequality. Since $1 > -\frac{9}{4}$, the answer choice $1$ does not satisfy the inequality.

E.

We are not given a specific value for answer choice E, so we cannot check if it satisfies the inequality.

Conclusion

In conclusion, the values of $c$ that will cause the quadratic equation $-x^2 + 3x + c = 0$ to have no real number solutions are $c < -\frac{9}{4}$. The answer choices that satisfy this inequality are $-5$ and $-\frac{9}{2}$.