Problem 135 - College Algebra

Question

Exercises \(134-136\) will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

Step-by-Step Solution

Answer

The solution to the equation \(x(x - 7) = 3\) is \(x = \frac{7 \pm \sqrt{61}}{2}\). This means there are essentially two solutions - \(x_1 = \frac{7 + \sqrt{61}}{2}\) and \(x_2 = \frac{7 - \sqrt{61}}{2}\)

Step 1: Rearrange the Equation

Re-arrange the equation in the form of \[ax^2 + bx + c = 0\]. In order to do so, expand the brackets and subtract 3 from both sides:\(x^2 - 7x - 3 = 0\)

Step 2: Apply the Quadratic Formula

Apply the quadratic formula, which is \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, a = 1, b = -7, and c = -3. Substituting the values in, you get:\[x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 * 1 * -3}}{2*1}\]Simplifying it, you get:\[x = \frac{7 \pm \sqrt{49+12}}{2}\]This simplifies to:\[x = \frac{7 \pm \sqrt{61}}{2}\]

Step 3: Calculate the Roots

Calculate the roots by evaluating both the plus and the minus in the solution for x. This leaves:1. \(x = \frac{7 + \sqrt{61}}{2}\)2. \(x = \frac{7 - \sqrt{61}}{2}\)These are the roots of given quadratic equation.

Key Concepts

Quadratic Formula

The quadratic formula is a cornerstone in solving quadratic equations, often seen in college algebra. Given a quadratic equation in the standard form \(ax^2 + bx + c = 0\), the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can find the roots of the equation. It's a versatile tool because it works for any quadratic equation, regardless of whether the roots are real or complex numbers. When applying it, ensure that you correctly identify values for \(a\), \(b\), and \(c\), substitute them into the formula, and perform the arithmetic to discover the equation's roots.

To prevent common errors, remember that the square root in the formula refers to both the positive and negative square roots of the discriminant \(b^2 - 4ac\). So when you solve, ensure to consider the \(\pm\) sign, which leads to two different solutions, corresponding to the two roots of the quadratic equation.

Factoring Quadratics

Factoring quadratics is another strategy to solve quadratic equations and is based on the principle that a product of zero implies that at least one of the factors must be zero. To factor a quadratic, express it as \(x^2 + bx + c = (x - p)(x - q) = 0\) where \(p\) and \(q\) are the roots. If you can find factors of \(c\) whose sum is \(b\), the quadratic can be factored easily. However, if factoring seems challenging, the quadratic formula remains a reliable fallback.

It's worth noting that sometimes, it's necessary to rearrange the equation by moving all terms to one side so that the other side equals zero. This form of equation is necessary for both factoring and using the quadratic formula effectively.

College Algebra

College algebra involves a variety of topics, with a significant emphasis on functions, their properties, and types of equations, which include linear, polynomial, and of course, quadratic equations. Understanding how to solve quadratic equations using multiple methods, such as graphing, factoring, and the quadratic formula, is essential. It's a subject that offers foundational skills necessary for advanced studies in fields such as engineering, computer science, and economics. A deep understanding of algebraic structures and the ability to manipulate algebraic expressions are critical competencies developed in this course.

Students should master both the theoretical aspects of algebra and the practical techniques for solving equations. Practicing with exercises, and understanding the logic behind each step, helps to solidify this comprehension.

Quadratic Equation Roots

Quadratic equation roots are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be real or complex and can be distinct or the same (in the case of a perfect square trinomial). Importantly, the discriminant \(b^2 - 4ac\) determines the nature of the roots: a positive discriminant indicates two distinct real roots, zero means one repeated real root, and a negative discriminant points to a pair of complex conjugate roots.

The roots represent where the quadratic function intersects the x-axis on a graph. Understanding the relationship between the discriminant and the roots is useful in graphing quadratic functions and predicting the number and type of solutions without actually calculating them.