Problem 28 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(2 x^{2}+6 x-3=0\)

Step-by-Step Solution

Answer

The exact solutions are \(x = \frac{-3 + \sqrt{15}}{2}\) and \(x = \frac{-3 - \sqrt{15}}{2}\).

Step 1: Identify the Type of Equation and Choose the Method

The equation given is a quadratic equation in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 2\), \(b = 6\), and \(c = -3\). We'll use the quadratic formula method to solve it, as it always provides the solutions for quadratic equations.

Step 2: Write the Quadratic Formula

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\] Substitute the values of \(a\), \(b\), and \(c\) into the formula.

Step 3: Calculate the Discriminant

First, calculate the discriminant \(b^2 - 4ac\): \[b^2 - 4ac = 6^2 - 4 \times 2 \times (-3).\]Simplify to find the value:\[36 + 24 = 60.\]The discriminant is 60.

Step 4: Substitute into the Quadratic Formula

Now substitute \(b = 6\), \(a = 2\), and \(\sqrt{60}\) into the quadratic formula:\[x = \frac{-6 \pm \sqrt{60}}{4}.\] Simplify the square root.

Step 5: Simplify the Square Root and the Equation

We know that \(\sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}\). Substitute back:\[x = \frac{-6 \pm 2\sqrt{15}}{4}.\] Simplify further by dividing the numerator and the denominator by 2:\[x = \frac{-3 \pm \sqrt{15}}{2}.\] This represents the two exact solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The formula itself is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula provides the solutions for any quadratic equation, regardless of its complexity, making it an essential technique. When using the quadratic formula, you need to identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. These values are then substituted directly into the formula.
This method not only finds solutions where the discriminant is non-negative, but it also encompasses scenarios where the equation might have complex solutions. Once you've plugged in the values, you simplify to find the roots of the equation.

Discriminant

In the quadratic formula, the discriminant is the part under the square root sign, specifically \(b^2 - 4ac\). The discriminant is a key component because it gives us important information about the nature of the roots of the quadratic equation. Here’s what the value of the discriminant can tell you:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also known as a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are complex and appear in conjugate pairs.

For the equation \(2x^2 + 6x - 3 = 0\), the discriminant is 60. Since 60 is greater than zero, it indicates two distinct real roots. Calculating the discriminant is a crucial step as it directs us towards what kind of solutions we might expect.

Exact Solutions

Exact solutions refer to finding the roots of an equation in their simplest, most precise form. In the context of quadratic equations, these solutions are typically expressed in terms of radicals or fractions rather than decimal approximations. To find the exact solutions of a quadratic equation, you substitute the values \(a, b,\) and \(c\) into the quadratic formula and perform the necessary algebraic manipulations.
For example, in our equation \(2x^2 + 6x - 3 = 0\), we use the quadratic formula to find:\[x = \frac{-3 \pm \sqrt{15}}{2}.\]
These are the exact roots of the equation. Attempting to substitute values or break down to decimals is useful in understanding approximate locations, but exact solutions retain all details and precision. This approach is particularly beneficial in mathematical proofs or when further exact calculations are required.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This format is crucial because it allows us to apply methods like factoring, completing the square, or the quadratic formula easily. It consists of:

• The quadratic term \(ax^2\);
• The linear term \(bx\);
• The constant term \(c\).

Each term plays a role in the behavior and graph of the quadratic equation. The coefficient \(a\) affects the parabola's width and direction, \(b\) influences the slope around the vertex, and \(c\) determines the point where the parabola crosses the y-axis.
When given any quadratic equation, like \(2x^2 + 6x - 3 = 0\), rewriting it in standard form is the first step before applying solution techniques such as the quadratic formula. Recognizing this structure helps in organizing your approach to solve the equation effectively.