Problem 48 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-3 x^{2}+2 x-1=0$$

Step-by-Step Solution

Answer

The quadratic equation \(-3x^2 + 2x - 1 = 0\) has no real solutions as the discriminant is negative.

Step 1: Identify a, b, and c from the quadratic equation

In a quadratic equation of the form \(ax^2 + bx + c = 0\), comparing it with the given equation \(-3x^2 + 2x - 1 = 0\), it can be seen that \(a = -3\), \(b = 2\), and \(c = -1\).

Step 2: Compute the discriminant

The discriminant is calculated as \(b^2 - 4ac\). Substituting in the values, we get \(Discriminant = (2)^2 - 4*(-3)*(-1) = 4 - 12 = -8\).

Step 3: Check the value of the discriminant

As the Discriminant is negative (-8), it implies that the given quadratic equation has no real solutions as square root of a negative number is not a real number.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation of degree 2, typically in the form of \(ax^2 + bx + c = 0\). These equations are called 'quadratic' because "quadratus" is the Latin word for 'square', relating to the square of the variable \( x^2 \). The general setup consists of:

• \( a \): the coefficient of \( x^2 \), which determines the parabola's openness and direction.
• \( b \): the coefficient of \( x \), affecting the parabola's symmetry axis.
• \( c \): the constant term, giving the vertical shift from the origin.

By comparing the given quadratic equation, \(-3x^2 + 2x - 1 = 0\), we identify \(a = -3\), \(b = 2\), and \(c = -1\). Solving quadratic equations often involves methods like factoring, completing the square, or using the quadratic formula, where the quadratic formula is useful when others are not readily applicable.

Role of the Discriminant

The discriminant is a crucial part of the quadratic formula, denoted as \(b^2 - 4ac\). It tells you about the number and types of solutions a quadratic equation may have. Specifically, it provides insight into whether solutions are real or imaginary. For any given quadratic equation, you compute the discriminant using \(a\), \(b\), and \(c\) values.

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there's exactly one real solution (a repeated root).
• If \(b^2 - 4ac < 0\), there are no real solutions, only two complex (imaginary) ones.

The provided equation \(-3x^2 + 2x - 1 = 0\) results in a discriminant of \(-8\), which is less than zero, indicating there are no real solutions for this equation.

Real Solutions in Quadratic Equations

Real solutions are values of \(x\) that satisfy the quadratic equation and form real numbers on the number line. They are derived from solving the quadratic equation using methods like the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Real solutions occur as pairs or single repeated roots:

• Two real and distinct solutions arise when the discriminant is positive.
• One real repeated solution exists when the discriminant is zero.
• No real solutions are present when the discriminant is negative, as the results involve the square root of a negative number, creating complex numbers.

In the problem given, after calculating the discriminant as negative (-8), it confirms that our quadratic equation \(-3x^2 + 2x - 1 = 0\) has no real number solutions.