Problem 53 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$-0.75 x^{2}+2=2 x$$

Step-by-Step Solution

Answer

The real solutions to the quadratic equation are \(x = 1.513\) and \(x = 1.754\)

Step 1: Rearrange the equation

Rearrange the equation in standard form by moving all terms to one side. So the equation becomes: \(-0.75x^2 - 2x + 2 = 0\)

Step 2: Identify a, b, and c

Identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = -0.75\), \(b = -2\), and \(c = 2\)

Step 3: Apply the quadratic formula

Apply the quadratic formula which is \(x = \[-b \pm \sqrt{(b^2-4ac)}/(2a)\}\). With \(a = -0.75\), \(b = -2\), and \(c = 2\), we substitute the values into the formula and solve for \(x\) to obtain the real solutions

Step 4: Solve for x

On performing the operations, we get \(x = 1.513\) and \(x = 1.754\). These are the solutions to the quadratic equation

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and \(x\) represents the variable. The formula, given by \(x = \frac{-b \(pm\) \sqrt{b^2 - 4ac}}{2a}\), is a universal solution that calculates the roots of any quadratic equation. When students encounter a quadratic equation, like \( -0.75x^2 - 2x + 2 = 0\), they can apply this formula to find the real solutions.
For easy recall, remember that the formula has three main components: the opposite of \(b\), the square root of the discriminant \(b^2 - 4ac\), and twice the coefficient \(a\). These values interact within the formula to unveil the real values of \(x\) that satisfy the quadratic equation in question.

Real Solutions

Real solutions of a quadratic equation refer to the values of \(x\) that are real numbers, which occur when the discriminant \(b^2 - 4ac\) is non-negative. When solving an equation like \( -0.75x^2 - 2x + 2 = 0\), we must check the discriminant to predict the nature of solutions. If \(b^2 - 4ac > 0\), there are two distinct real solutions. If \(b^2 - 4ac = 0\), there is one real solution, often called a 'repeated root'. In case \(b^2 - 4ac < 0\), there are no real solutions, instead, there are two complex solutions.
In the exercise at hand, since the calculated values of \(x\) are 1.513 and 1.754, we can confirm that the discriminant was non-negative and thus the equation had two real solutions. Understanding the discriminant not only helps in finding real solutions but also in understanding the characteristics of the quadratic equation.

Solving Quadratic Equations

Solving quadratic equations is a foundational skill in algebra. The process involves manipulating the equation to find the values of \(x\) that make the equation true. To solve an equation like \( -0.75x^2 - 2x + 2 = 0\), the first step is to write it in standard form, as was done in Step 1 of the solution. Identifying \(a\), \(b\), and \(c\) is crucial as these coefficients are plugged into the quadratic formula.
After applying the formula, you often obtain two solutions, as quadratic equations typically have up to two real roots. In the given exercise, the students calculated two real solutions, 1.513 and 1.754, after substituting the identified coefficients into the quadratic formula. Through practice, solving quadratic equations becomes an intuitive process that strengthens a student's problem-solving skills in algebra.