Problem 79 - Algebra 1: Concepts and Skills

Question

Tell whether the equation has two solutions, one solution, or no real solution. $$ x^{2}-2 x-24=0 $$

Step-by-Step Solution

Answer

The equation \(x^2 - 2x - 24 = 0\) has two real and distinct solutions.

Step 1 Identification

This is a quadratic equation in the form \(ax^2 + bx + c=0\). Here, \(a=1\), \(b=-2\), and \(c=-24\)

Step 2: Calculate the Discriminant

The discriminant is given by \(b^2 - 4ac\). So it is: \((-2)^2 - 4*1*(-24) = 4 + 96 = 100\)

Step 3: Determine the Number of Real Solutions

Since the discriminant is greater than 0, the equation has two real and distinct solutions.

Key Concepts

Discriminant

In quadratic equations, the discriminant is a key component that tells us about the nature of the roots of the equation. It is calculated from the coefficients of the quadratic equation using the formula given by \(b^2 - 4ac\). In our example, the quadratic equation \(x^2 - 2x - 24 = 0\) has coefficients \(a = 1\), \(b = -2\), and \(c = -24\).

• First, we compute \((-2)^2 = 4\).
• Then, compute \(4 \times 1 \times (-24) = -96\).
• Finally, combine these to get the discriminant: \(4 - (-96) = 100\).

Since we're interested in understanding the nature of the solutions, the discriminant's value gives crucial information. Depending on its value, it will tell us how many real solutions the equation has.

Real Solutions

The number of real solutions a quadratic equation has is determined by its discriminant:

• If the discriminant is positive (greater than zero), like in our example where it is 100, the equation has two distinct real solutions. This means the parabola represented by the equation intersects the x-axis at two points.
• If the discriminant equals zero, there is exactly one real solution, meaning the parabola touches the x-axis at just one point, known as a double root.
• If the discriminant is negative, there are no real solutions. Instead, the solutions are complex numbers, and the parabola does not intersect the x-axis at all.

For the equation \(x^2 - 2x - 24 = 0\), the discriminant is 100, leading us to conclude that there are two real and distinct solutions.

Quadratic Formula

The quadratic formula is a method used to find the solutions of a quadratic equation. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula will work for any quadratic equation of the form \(ax^2 + bx + c = 0\). Knowing the discriminant, we can plug the values directly into the quadratic formula.

• For our equation, \(a = 1\), \(b = -2\), and \(c = -24\).
• The discriminant \(b^2 - 4ac = 100\), so we have \(\sqrt{100} = 10\).
• Plugging these into the formula gives us: \(x = \frac{-(-2) \pm 10}{2 \times 1} = \frac{2 \pm 10}{2}\).
• This will yield the two solutions: \(x = \frac{12}{2} = 6\) and \(x = \frac{-8}{2} = -4\).

Thus, by using the quadratic formula, we find the specific real solutions for \(x\) in the equation \(x^2 - 2x - 24 = 0\).