Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 2+2 x-x^{2}=0 $$

Step-by-Step Solution

Answer

The solutions of the quadratic equation \(2+2x-x^2=0\) are \(x = 1 - \sqrt{3}\) and \(x = 1 + \sqrt{3}\)

Step 1: Rearrange the quadratic equation to standard form

Put the quadratic equation into the standard form \(ax^2 + bx + c = 0\). The given equation is \(2+2x-x^2=0\). Rearranging this equation gives \(-x^2+ 2x +2 = 0\)

Step 2: Identify values of a, b, and c

In the equation \(-x^2+ 2x + 2 = 0\), a is -1, b is 2, and c is 2.

Step 3: Substitute these values into the Quadratic Formula and solve for x

Substitute a = -1, b = 2, and c = 2 into the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to get: \[x = \frac{-2 \pm \sqrt{(2)^2 - 4*(-1)*2}}{2*(-1)}\] This simplifies to: \[x = \frac{-2 \pm \sqrt{4+8}}{-2}\] which further simplifies to: \[x = \frac{-2 \pm \sqrt{12}}{-2}\] Finally, we get the solutions: \[x = \frac{-2 + \sqrt{12}}{-2} \quad \text{and} \quad x = \frac{-2 - \sqrt{12}}{-2}\]

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means it features an \(x^2\) term as its highest-order term. This type of equation usually appears in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Because of its quadratic nature, it can produce up to two solutions or roots. The solutions to a quadratic equation can be real or complex. Real solutions can be:

• Distinct: Two different numbers.
• Repeated: The same number counted twice.

Complex solutions occur in pairs and come about when the equation does not cross the x-axis. Quadratic equations are fundamental in various fields, including physics, engineering, and economics, because they often describe natural phenomena, such as the trajectory of an object in motion.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\). This format helps in applying methods like the Quadratic Formula to find solutions efficiently.To write any quadratic equation in standard form, you should:

• Ensure the equation equals zero: Move all terms to one side and have '0' on the other.
• Order terms by decreasing power: The \(x^2\) term should come first, followed by the \(x\) term, and lastly the constant term.

In the exercise example, the equation \(2 + 2x - x^2 = 0\) needs rearranging. Reordering gives us \(-x^2 + 2x + 2 = 0\), but to follow the better practice, it's often reordered as \(x^2 - 2x -2 = 0\). The standard form simplifies using various solving strategies, making it versatile for solving problems effectively.

Solving Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. One of the most reliable methods is using the Quadratic Formula, which is represented by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the solutions directly using the coefficients \(a\), \(b\), and \(c\) from the standard form. The steps to apply the Quadratic Formula include:

• Identify \(a\), \(b\), and \(c\) from your equation.
• Substitute these values into the formula.
• Solve the expression under the square root, known as the discriminant (\(b^2 - 4ac\)).
• Consider the cases depending on the discriminant:
• If \(b^2 - 4ac > 0\), you have two real and distinct solutions.
• If \(b^2 - 4ac = 0\), you get a repeated solution.
• If \(b^2 - 4ac < 0\), the solutions are complex.

In our worked example, \(b^2 - 4ac\) yielded a positive value, indicating two distinct real solutions. Mastering these techniques is crucial for solving any quadratic equation methodically.