Problem 30 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. \(-5 x^{2}+6 x-6=0\)

Step-by-Step Solution

Answer

The quadratic equation \(-5x^{2} + 6x - 6 = 0\) has no real solutions.

Step 1: Identify Coefficients

From the equation \(-5x^{2} + 6x - 6 = 0\), identify the coefficients a, b, and c. In this instance, \(a=-5\), \(b=6\), and \(c=-6\).

Step 2: Apply to Discriminant

Apply the values of a, b, and c to the discriminant formula \(b^{2}-4ac\). This gives \(6^{2} -4*(-5)*(-6)\)

Step 3: Calculate Discriminant

Calculate the value to get the discriminant. This is \(36 - 120 = -84\)

Step 4: Determine the Number of Real Solutions

As the value of the discriminant is less than zero, it means the equation \(-5x^{2} + 6x - 6 = 0\) has no real solutions.

Key Concepts

Discriminant in Quadratics

The discriminant is a powerful tool in determining the nature of the solutions for quadratic equations. In a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, the discriminant is given by the formula \(D = b^2 - 4ac\).

The value of the discriminant can tell us a lot about the roots of the equation:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, which is also called a repeated or double root.
• If \(D < 0\), there are no real solutions; instead, the solutions are complex and conjugate to each other.

In our example \(-5x^2 + 6x - 6 = 0\), we calculated the discriminant to be \(-84\), which is less than zero. Thus, we can conclude that this quadratic equation has no real solutions, and its solutions are complex numbers.

Identifying Coefficients

To work with any quadratic equation effectively, recognizing the coefficients is fundamental. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where

• \(a\) is the coefficient of the quadratic term \(x^2\),
• \(b\) is the coefficient of the linear term \(x\), and
• \(c\) is the constant term.

Identifying these coefficients is a precursor to applying various methods for solving the equation, including finding the roots by factoring, completing the square, using the quadratic formula, or determining the discriminant.

In the equation \(-5x^2 + 6x - 6 = 0\), we identify that \(a = -5\), \(b = 6\), and \(c = -6\). These values are essential for various computations, as seen with the discriminant in the previous section.

Real Solutions of Quadratic Equations

Understanding the conditions for the existence of real solutions in a quadratic equation helps in anticipating the nature of the roots before even solving the equation. As we discussed with the discriminant, a quadratic equation will have:

• Two real solutions if the discriminant is positive,
• One real solution if the discriminant is zero, and
• No real solutions if the discriminant is negative.

When a quadratic equation has no real solutions, it means you cannot find real number values for \(x\) that satisfy the equation. Instead, the solutions involve imaginary numbers (involving the square root of a negative number).

For instance, with our example equation \(-5x^2 + 6x - 6 = 0\), since the discriminant is negative, we affirm there are no real solutions. In such cases, the solutions can still be found using the quadratic formula, resulting in complex numbers which have real and imaginary components.