Question

Given that the equation \(3 x^2+2 x-8=0\) has two distinct solutions, what is the value of the smaller solution subtracted from the larger solution?

Short answer

The difference between the two solutions is \(\frac{10}{3}\).

Step-by-step solution

Find the Discriminant

Using the quadratic equation, \(ax^2+bx+c=0\), the discriminant is defined as: \(D=b^2-4ac\). Substituting the given values from the equation \(3x^2+2x-8=0\), we get: \(a=3, b=2, c=-8\) Discriminant, \(D = (2)^2 - 4(3)(-8) = 4 + 96 = 100\) Since the discriminant is greater than zero, we confirm that the quadratic equation has two distinct solutions.

Find the Solutions

Now let's use the quadratic formula to find the two solutions. The quadratic formula states that: \(x = \frac{-b \pm \sqrt{D}}{2a}\) Using the values we have, we get: \(x = \frac{-2 \pm \sqrt{100}}{2(3)}\) Now, let's calculate the two solutions: \(x_1 = \frac{-2 + 10}{6} = \frac{8}{6} = \frac{4}{3}\) \(x_2 = \frac{-2 - 10}{6} = \frac{-12}{6} = -2\)

Calculate the Difference

Now that we have the two solutions, let's find the difference between the larger solution and the smaller solution: Difference = Larger solution - Smaller solution = \(\frac{4}{3} - (-2)\) Adding the smaller solution to the larger solution, we get: Difference = \(\frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3}\) Thus, the difference between the two solutions is \(\frac{10}{3}\).

Key concepts

Discriminant of a Quadratic Equation

Understanding the discriminant is crucial when tackling quadratic equations. It's the part under the square root in the quadratic formula: \(b^2 - 4ac\). This value tells us the nature of the solutions to a quadratic equation \(ax^2 + bx + c = 0\).

Here's what the discriminant indicates:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution (a repeated root).
• If \(D < 0\), there are no real solutions, only complex ones.

For the exercise \(3x^2+2x-8=0\), we calculated the discriminant to be 100, which means there are two distinct real solutions. The discriminant solves the initial query even before finding the actual roots and simplifies the process, especially in time-sensitive exams like the SAT.

Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) is a reliable method to find the solutions of any quadratic equation. Tackle it in two parts: first, calculate the discriminant \(D\), then plug values into the formula.

The \pm symbol means we will perform the calculation twice: once with a plus to find the larger solution (\(x_1\)) and once with a minus for the smaller one (\(x_2\)). This method ensures you'll find both solutions quickly and accurately. It's handy in many areas, from algebra class to standardized tests like the SAT.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra. There are several methods, including factoring, completing the square, graphing, and using the quadratic formula, which is often the most straightforward approach.

To solve using the quadratic formula, follow these steps:

• Identify \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\).
• Calculate the discriminant \(D\).
• Substitute \(a\), \(b\), and \(D\) into the quadratic formula.
• Simplify to find the solutions \(x_1\) and \(x_2\).

In our example, we found the solutions \(\frac{4}{3}\) and \(\frac{-12}{6}\) by following these steps. This systematic approach helps ensure accuracy and is a vital skill for any math student.

SAT Math Preparation

Preparing for the math section of the SAT involves understanding concepts like solving quadratic equations. It's not just about memorizing formulas but also knowing when and how to apply them efficiently.

Practice is essential. Here's how you can prepare:

• Start early and establish a study schedule.
• Understand and practice different methods to solve quadratic equations.
• Focus on accuracy and speed by timing yourself on practice problems.
• Review mistakes to prevent them in the future.

Working through problems like the one we've discussed not only builds your quadratic equation-solving skills but also prepares you for the types of algebra questions you'll encounter on the SAT.