Problem 80 - College Algebra

Question

Solve each equation in Exercises \(73-98\) by the method of your choice. \(2 x^{2}+3 x=1\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -3/4 + sqrt(17)/4\) and \(x = -3/4 - sqrt(17)/4\).

Step 1: Identify a, b and c

Start by writing the equation in general form, \(ax²+bx+c=0\). Here, \(a=2\), \(b=3\) and \(c=-1\).

Step 2: Calculate the Discriminant

Calculate the discriminant \(b²-4ac\) which is used in the quadratic formula. For this equation, the discriminant will be \(3² - 4*2*(-1)\) which equals \(9 + 8 = 17\).

Step 3: Apply the Quadratic Formula

According to the quadratic formula, the roots of the equation are given by \(x = [-b ± sqrt(discriminant)] / 2a\). Substitute \(a=2\), \(b=3\) and \(discriminant=17\) into this formula to get: \(x = [-3 ± sqrt(17)] / 4\). this will give two answers one using plus operation and the other one using the minus operation. Thus our roots are \(x = -3/4 + sqrt(17)/4\) and \(x = -3/4 - sqrt(17)/4\).

Key Concepts

Quadratic Formula

The quadratic formula is a valuable tool for solving quadratic equations, which are equations of the second degree, meaning they include an x^2 term. This powerful formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). To use it, one must identify the coefficients a, b, and c from the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \).

In our exercise, the given equation \( 2x^2 + 3x = 1 \) is first rewritten into the standard form by bringing all terms to one side, yielding \( 2x^2 + 3x - 1 = 0 \). Here, a=2, b=3, and c=-1. Using these values, the quadratic formula is applied to find the roots of the equation, which are the values of x that satisfy the equation.

Applying the formula offers a systematic way to find both real and complex solutions, streamlining the process of solving any quadratic equation.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a component of the quadratic formula that determines the nature and number of roots of the equation. It is represented by \( b^2 - 4ac \) and sits under the square root in the quadratic formula.

In the context of our solved equation, we calculate the discriminant as follows: \({3}^2 - 4 \cdot 2 \cdot (-1)\), which simplifies to 17.

The value of the discriminant can tell us whether the roots of the quadratic equation are real and distinct, real and identical, or complex:

• If the discriminant is positive (> 0), there are two distinct real roots.
• If the discriminant is zero (0), there is exactly one real root (also called a repeated or double root).
• If the discriminant is negative (< 0), there are two complex roots.

In our example, the discriminant is positive, indicating that the equation has two distinct real roots as we found.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions to the equation; they are the values of x that make the equation true. In other words, they are the x-intercepts of the quadratic function on a graph.

For the quadratic equation in question, using the values derived from the quadratic formula and our discriminant (17), we find the roots to be \( x = -\frac{3}{4} + \frac{\sqrt{17}}{4} \) and \( x = -\frac{3}{4} - \frac{\sqrt{17}}{4} \). These roots are unique to this quadratic equation.

Knowing the roots is especially useful in graphing the quadratic function and in various applications such as physics, economics, and engineering, where finding the maximum or minimum value of a quadratic function is often necessary. The ease of solving for the roots with the quadratic formula showcases its utility in both simple and complex real-world problems.