Problem 71 - Algebra 2

Question

Use the discriminant to find the number of real solutions. $$ -x^{2}+2 x-8=0 $$

Step-by-Step Solution

Answer

The discriminant is -28, indicating there are no real solutions.

Step 1: Write down the quadratic formula's discriminant

For a quadratic equation in the form of ax^2 + bx + c = 0, the discriminant is given by the formula \(\text{Discriminant, } D = b^2-4ac\). The discriminant helps determine the nature of the roots.

Step 2: Identify the coefficients

For the given equation \( -x^2 + 2x - 8 = 0 \), identify the coefficients: \(\text{a} = -1\), \(\text{b} = 2\), and \(\text{c} = -8\).

Step 3: Substitute the coefficients into the discriminant formula

Substitute a, b, and c into the discriminant formula to find D: \(\text{D} = (2)^2 - 4(-1)(-8)\).

Step 4: Calculate the discriminant

Calculate the value of the discriminant: \(\text{D} = 4 - 4(-1)(-8) = 4 - 32 = -28\). Since the discriminant is negative, this means there are no real solutions.

Key Concepts

Number of Real Solutions

Understanding the number of real solutions to a quadratic equation is made simpler by using the discriminant, a vital concept in algebra. The discriminant is a specific value obtained from the coefficients of the quadratic equation of the form \(ax^2 + bx + c = 0\). It tells us whether the equation has two distinct real solutions, one real solution, or no real solutions at all.

The formula to find the discriminant is \(D = b^2 - 4ac\). By plugging the values into this formula, you can determine the nature of solutions without actually solving the equation. Keep in mind the following conditions:

Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) is a reliable method used to solve quadratic equations. The term under the square root, \(b^2 - 4ac\), is the discriminant we discussed earlier. This formula provides the exact solutions for \(x\), given that a, b, and c are known from the equation's standard form.

To use the quadratic formula effectively, ensure the equation is first set to zero, aligning with the form \(ax^2 + bx + c = 0\). The plus-minus symbol, \(\pm\), indicates that you'll get two possible values for \(x\), which correlates with the potential for two real solutions. However, when the discriminant is zero or negative, you'll get one or no real solution, respectively.

Nature of the Roots

The nature of the roots of a quadratic equation is fundamentally connected to the discriminant's value. When the discriminant is positive (\(D>0\)), there are two distinct real roots. If the discriminant equals zero (\(D=0\)), this results in a single, repeated real root, because both solutions given by the quadratic formula converge into one. However, a negative discriminant (\(D<0\)), which we encountered in our example problem, implies that there are no real roots; instead, the solutions are complex numbers.

In summary, the discriminant not only gives us the number of real solutions but also provides insight into their nature—whether they are distinct, repeated, or complex.