Problem 33 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. (Lesson 9.7) $$ x^{2}-14 x+49=0 $$

Step-by-Step Solution

Answer

The given equation has one solution.

Step 1: Identify Coefficients

In the equation \(x^{2}-14x+49=0\), \(a = 1\), \(b = -14\), and \(c = 49\).

Step 2: Calculate the Discriminant

The discriminant is given by the formula \(b^{2}-4ac\). Therefore, substituting the values of the coefficients a, b and c into the discriminant formula gives \((-14)^{2}-4*1*49 = 0\).

Step 3: Interpret the Discriminant

Since the discriminant equals 0, it shows that this equation has one solution.

Key Concepts

Discriminant in Quadratic Equations

In understanding quadratic equations, the discriminant plays a pivotal role. What is this discriminant, you ask? Simply put, it's a part of the quadratic formula that provides critical information about the nature of the solutions to the equation. Specifically, the discriminant is expressed as the part under the square root in the quadratic formula, calculated as \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).

The value of the discriminant reveals:

• If it's positive, you'll find two distinct real solutions.
• If it's zero, the equation has exactly one real solution.
• If it's negative, no real solutions exist, but there are two complex solutions instead.

The discriminant gives us a quick and efficient way to determine not just the number, but also the types of solutions without actually solving the equation. It's like a forecast that tells us what to expect before we proceed with further calculations.

Finding Quadratic Equation Solutions

Finding solutions to quadratic equations can be akin to solving a puzzle – it requires a mixture of knowledge and process. Once we know the discriminant, as described earlier, we have an idea of how many solutions to expect. But how exactly do we find those solutions? There are various methods, such as factoring, completing the square, or graphing. However, the most widely used method is the application of the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\)Here's the process:

1. Identify the coefficients a, b, and c from the quadratic equation \(ax^2 + bx + c = 0\).
2. Plug these values into the quadratic formula.
3. Simplify the calculations under the square root and the fraction to find the value(s) of x, which are the solutions.

Using this approach, you can solve any quadratic equation providing it has real solutions. The critical part is careful arithmetic to ensure the right solutions.

Quadratic Formula

The quadratic formula is the ace up the sleeve for solving quadratic equations. It's a sure-fire method that works every time for finding the solutions of \(ax^2 + bx + c = 0\). The formula is: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).

The magic lies in the 'plus-minus' symbol (\(\pm\)), indicating that there are generally two solutions to the equation. This symbol leads us to two different values of \(x\) upon solving. The term \(\sqrt{{b^2 - 4ac}}\) is the discriminant which can simplify to a positive number, zero, or a negative number, thus determining the realness or complexity of the solutions.

When the quadratic formula is applied to our example equation \(x^2 - 14x + 49 = 0\), we calculate the discriminant as zero. According to the formula, this means that there's only one real solution, because the square root of zero is zero, which nullifies the effect of the plus-minus symbol, leaving us with a single solution for \(x\). The quadratic formula thus serves as a universal tool, ensuring that a solution is always within reach.