Problem 45 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by using the quadratic formula. Find only real solutions. $$3-x-x^{2}=0$$

Step-by-Step Solution

Answer

The real solutions to the quadratic equation \(3-x-x^{2}=0\) are \(x1 = (-1 + \sqrt{13})/2\) and \(x2 = (-1 - \sqrt{13})/2\).

Step 1: Standard Form for Quadratic Equation

Firstly, convert the equation to the standard form. By rearranging terms and signs in the given equation, the equation is written as \(x^{2} + x - 3 = 0\) where \(a = 1\), \(b = 1\), and \(c = -3\).

Step 2: Applying Quadratic Formula

Next, apply the quadratic formula \(x = [-b \pm \sqrt{(b^{2}-4ac)}]/(2a)\). Substituting the values of \(a\), \(b\), and \(c\) for this formula, the formula becomes \(x = [-1 \pm \sqrt{(1^{2}-4*1*-3)}]/(2*1) = [-1 \pm \sqrt{13}]/2.\)

Step 3: Finding Real Solutions

Lastly, find the real solutions by simplifying the expressions from step 2. The two real solutions are \(x1 = (-1 + \sqrt{13})/2\) and \(x2 = (-1 - \sqrt{13})/2\).

Key Concepts

Real Solutions

When solving a quadratic equation using the quadratic formula, you're often looking for what are called "real solutions". Real solutions refer to the set of solutions that do not involve imaginary numbers. Imaginary numbers arise when you take the square root of a negative number because there is no real number whose square is negative.

In terms of the quadratic formula, whether or not a quadratic equation has real solutions depends on the discriminant, which is the expression under the square root: \(b^2 - 4ac\). When this value is:

• Positive: The equation has two distinct real solutions.
• Zero: The equation has exactly one real solution, sometimes called a double root.
• Negative: The equation has no real solutions but two complex solutions.

In our problem, for the equation \(3 - x - x^2 = 0\) which we rearranged to \(x^2 + x - 3 = 0\), the discriminant can be computed as \(1^2 - 4(1)(-3)\). This simplifies to \(13\), a positive number, indicating that there are two real solutions.

Standard Form

The standard form of a quadratic equation is essential for solving the equation using the quadratic formula. A quadratic equation in standard form is presented as \(ax^2 + bx + c = 0\). Having the equation in this form makes it straightforward to identify the coefficients \(a\), \(b\), and \(c\), which are required for the quadratic formula.

To convert the equation \(3 - x - x^2 = 0\) to standard form, rearrange the terms to organize them by decreasing powers of \(x\). This process involves:

• Bringing terms involving \(x\) to one side of the equation.
• Listing the \(x\) terms in order of decreasing exponent.
• Standardizing the equation to match the \(ax^2 + bx + c = 0\) form.

For our equation, after rearranging, we have \(x^2 + x - 3 = 0\), showing therefore, \(a = 1\), \(b = 1\), and \(c = -3\). This form is crucial for plugging the values into the quadratic formula.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, generally represented as \(ax^2 + bx + c = 0\). It is characterized by its highest exponent, the square of \(x\), making it a quadratic equation. Solving these equations is typically done using methods such as factoring, completing the square, or applying the quadratic formula.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a reliable method for finding the solutions of any quadratic equation, regardless of whether their factors are easily found. The variables \(a\), \(b\), and \(c\) correspond to the coefficients of the quadratic equation in standard form. By calculating the discriminant \(b^2 - 4ac\) and then applying the rest of the formula, you can find the real solutions of the equation.

In the problem \(3 - x - x^2 = 0\), which rearranges to \(x^2 + x - 3 = 0\), we use the quadratic formula to find the solutions. Here, substituting \(a = 1\), \(b = 1\), and \(c = -3\) into the formula helps us solve for both real roots of the equation: \(x_1 = \frac{-1 + \sqrt{13}}{2}\) and \(x_2 = \frac{-1 - \sqrt{13}}{2}\). These solutions are the outputs of applying the quadratic formula to the problem at hand.