Problem 27 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(6 x^{2}-2 x+4=0\)

Step-by-Step Solution

Answer

The equation \(6x^{2} - 2x +4=0\) has no real solutions.

Step 1: Identify the coefficients

The given equation is \(6x^{2} - 2x +4=0\). Here, \(a = 6\), \(b = -2\), and \(c = 4\).

Step 2: Substitute the coefficients into the determinant

The determinant of the equation is computed as \(b^{2}-4ac\). Substituting the values of a, b, and c gives \((-2)^{2} - 4 * 6 * 4\).

Step 3: Evaluate the determinant

Computing the above expression gives \(4 - 96 = -92\). Since the determinant is less than 0, the equation has no real solutions.

Key Concepts

Discriminant

The discriminant is a key element in understanding the nature of the roots of a quadratic equation. It is the part of the quadratic formula under the square root and is defined by the expression \( b^2 - 4ac \). The value of the discriminant tells us whether a quadratic equation has two real solutions, one real solution, or no real solutions at all. Specifically:

• If the discriminant is positive (\( > 0 \)), this indicates that the equation has two distinct real solutions.
• If the discriminant is zero (\( = 0 \)), it means that there is exactly one real solution, which is also referred to as a repeated or double root.
• Lastly, if the discriminant is negative (\( < 0 \)), the equation has no real solutions; instead, it has two complex conjugate solutions.

In our example, the discriminant \( (-2)^2 - 4 \times 6 \times 4 \) equals -92, which means there are no real solutions to the equation \(6x^{2} - 2x + 4 = 0\).

Quadratic Formula

The quadratic formula is a powerful tool that allows us to solve any quadratic equation, regardless of its coefficients. The formula is \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). It directly utilizes the coefficients of the quadratic equation (\(a\), \(b\), and \(c\)) and the discriminant (within the square root) to determine the equation's solutions. The '±' symbol indicates that there are typically two solutions represented by adding and subtracting the square root of the discriminant. When the discriminant is negative, as in our example, this means that the square root part of the formula will be of a negative number, which is not possible in the real number system and thus, we conclude no real solutions exist. Instead, complex solutions can be found, which include imaginary numbers.

Real Solutions of Quadratic Equations

Real solutions of quadratic equations are the x-values where the parabola, represented by the equation, crosses the x-axis. These intersection points are the roots or zeros of the quadratic function. For a quadratic equation in the standard form \(ax^2 + bx + c = 0\), whether we have real solutions or not depends on the discriminant (\(b^2 - 4ac\)). As discussed earlier, a negative discriminant implies no real solutions because you cannot take the square root of a negative number in the real number system. Since the discriminant of our equation is -92, a negative number, the graph of the parabola does not intersect the x-axis at any point. Thus, our equation \(6x^{2} - 2x + 4 = 0\) does not have real roots and does not cross the x-axis.

Coefficients of Quadratic Equation

The coefficients of a quadratic equation are the numerical factors that precede each term. In the equation of the form \(ax^2 + bx + c = 0\), \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term. These coefficients play a critical role in shaping the graph of the quadratic function and in determining the nature and number of solutions. For instance, the coefficient \(a\) determines the direction of the parabola (upwards when positive, downwards when negative), while \(b\) and \(c\) help define its position on the graph. In the given exercise, \(a = 6\), \(b = -2\), and \(c = 4\) which tells us the parabola opens upward, and its vertex is shifted from the origin. Knowing these coefficients helps us to apply the quadratic formula correctly and to understand the characteristics of the parabola derived from the equation.