Problem 51 - Algebra 1: Concepts and Skills

Question

For which value of \(c\) will \(-3 x^{2}+6 x+c=0\) not have a real solution? A) \(c<-3\) B) \(c=-3\) C) \(c>-3\) D) \(c=3\)

Step-by-Step Solution

Answer

The value of c for which the given quadratic equation will have no real solution is when \(c < -3\) which corresponds to answer choice A)

Step 1: Identifying the quadratic equation

The given equation is in the form of \(ax^2 + bx + c = 0 \) where a = -3, b = 6, c is unknown. Remember, the discriminant is \(b^2 - 4ac\). We want this to be less than zero for no real solutions.

Step 2: Substituting in the Discriminant

Substitute the values of a and b into the discriminant. The discriminant becomes \( (6)^2 - 4*(-3)*c \) which simplifies to \(36 + 12c\).

Step 3: Finding the value of c so the discriminant is less than zero

To ensure no real solutions, the discriminant needs to be less than zero . Set \(36 + 12c < 0\) and solve for c. Dividing by 12, we get \(c < -3\).

Key Concepts

Quadratic Equations

Quadratic equations are a staple in algebra, usually written in the standard form: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The characteristics of quadratic equations are:

• They graph as a parabola — either opening up or down depending on the sign of \( a \).
• The equation can have zero, one, or two real solutions.
• Solving these equations often involves factoring, completing the square, or using the quadratic formula.

Understanding the nature of their solutions requires exploring the discriminant and how it influences whether the solutions are real or complex.

Real Solutions

In the context of quadratic equations, real solutions refer to the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \) and are real numbers. The discriminant, denoted as \( \Delta = b^2 - 4ac \), is crucial in determining the nature of these solutions.

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution (also called a repeated or double root).
• If \( \Delta < 0 \), the solutions are not real; instead, they are complex.

In cases where real solutions are desired, ensuring that the discriminant is positive or zero is crucial. Alternatively, if the goal is to avoid real solutions, ensuring that the discriminant is negative will guarantee complex outcomes.

Inequalities

Inequalities involve mathematical statements indicating that one quantity is less than, greater than, less than or equal to, or greater than or equal to another. They are not just related to numbers but also extend to functions and equations. Solving inequalities often involves:

• Rearranging terms similar to equations but with care, particularly when multiplying or dividing by a negative number, which flips the direction of the inequality sign.
• Substituting values to test solution regions.
• Understanding context, as with the case discriminants defining solution types in quadratic equations.

For quadratic equations, solving inequalities often intersects with determining valid ranges for the discriminant, helping us identify when a given quadratic has solutions of a particular nature.