Problem 70 - Algebra 1: Concepts and Skills

Question

Consider the equation \(3 x^{2}-44=x^{2}+84\) Which statement is correct? A. The equation has exactly one solution. B. The equation has two solutions. C. The equation has no real solution. D. The number of solutions cannot be determined.

Step-by-Step Solution

Answer

B. The equation has two solutions.

Step 1: Simplify the equation

To start with, let's rearrange the equation to get a quadratic equation in the standard form \(ax^{2} + bx + c = 0\). Here, upon simplifying the given equation \(3 x^{2}-44=x^{2}+84\), we get \(2x^{2} - 128 = 0\). Thus, the equation's coefficients are a=2, b=0, c=-128.

Step 2: Calculate the discriminant

Next, let's calculate the discriminant (\(b^{2}-4ac\)) using the obtained coefficients which determines the number of real solutions. The discriminant in this case is: \(b^{2} - 4ac = (0)^{2} - 4*(2)*(-128) = 1024\).

Step 3: Conclude the number of solutions

As the calculated discriminant is greater than 0, it means the given quadratic equation has 2 real solutions.

Key Concepts

Discriminant in Quadratic Equations

When we encounter a quadratic equation, the first thing that might come to mind is how many solutions does it have? The discriminant is a key concept that provides us with an answer to this question.

The discriminant is found using the formula \( b^2 - 4ac \) where \( a, b, \) and \( c \) are the coefficients of the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). The value of the discriminant informs us about the nature and number of the solutions:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (also known as a repeated or double root).
• If the discriminant is negative, the equation has no real solutions but two complex solutions.

For instance, with the given problem, calculating the discriminant as \(0^2 - 4(2)(-128) = 1024\) indicates that there are two distinct real solutions. Understanding the discriminant is crucial for predicting the solutions even before you solve the equation fully.

Solving Quadratic Equations

Quadratic equations can be tackled through various methods, each with its perks. The most common techniques include factoring, completing the square, using the quadratic formula, and graphing.

When we solve by factoring, we look for two numbers that multiply to give \(ac\) and add to give \(b\). If the quadratic is factorable, this method is quick and efficient. However, not all quadratics are easily factorizable. When you complete the square, you rewrite the equation so that you have a perfect square trinomial, which you can then solve by taking the square root of both sides. The quadratic formula, derived from the process of completing the square, works for any quadratic equation and is \( x = \frac{{-b \(pm\) \sqrt{{b^2 - 4ac}}}}{{2a}} \). Graphing involves plotting the quadratic on a graph to find the points where it crosses the x-axis, which represent the solutions.

In our exercise, we could solve the simplified equation \(2x^2 - 128 = 0\) by factoring, but it's simpler here to isolate \(x^2\) and take the square root, leading to \(x = \(pm\) 8\), indicating two solutions, \( x = 8 \) and \( x = -8 \).

Standard Form of a Quadratic Equation

A quadratic equation may not always be presented to you in a neat little package ready for analysis. Therefore, the first task is often to rearrange it into the standard form, which is \( ax^2 + bx + c = 0 \) where \( a \) is not equal to zero. The a-term dictates the direction of the parabola's opening, the b-term relates to the symmetry and vertex of the parabola, and the c-term indicates the y-intercept when \( x = 0 \).

In our starter problem, we took the given equation \(3 x^2 - 44 = x^2 + 84\) and rearranged it to get \(2x^2 - 128 = 0\), which is in the standard form. By doing so, we can easily determine the coefficients \( a = 2, b = 0, \) and \( c = -128 \), which are pivotal in finding the solution by any means whether it is factoring, using the quadratic formula, or completing the square. This process simplifies the problem and creates a clear path for solving the quadratic equation.