Problem 50 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$3 x^{2}-6 x-3=12-6 x$$

Step-by-Step Solution

Answer

The solutions for x are -1 and 3.

Step 1: Simplify the Equation

First, rearrange the equation to bring all terms on one side. This gives \(3x^2-6x+3-12=-6x+6x\) which simplifies to \(3x^2-6x-9=0\).

Step 2: Factoring the quadratic equation

By factoring this equation, one can solve for x directly. The factored form of this equation is \((3x + 3)(x - 3)=0\).

Step 3: Solving for x

Setting each factor equal to zero gives possible solutions. So we get \(3x+3=0\) and \(x-3=0\). Solving for x in each equation gives x = -1 and x = 3 as the solutions.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method for solving quadratic equations, which are polynomials of the second degree, typically in the form of \(ax^2 + bx + c = 0\). To factor a quadratic, one looks for two binomials that multiply together to give the original quadratic equation.

In our exercise, the step was to transform the equation \(3x^2-6x-9=0\) into a product of two binomials. The key is to find two numbers that multiply to give the product of the quadratic coefficient (3) and the constant term (-9), which is -27, and also add up to the linear coefficient (-6). These numbers are 3 and -9, leading to the factors (3x + 3) and (x - 3).

Why Factoring Matters

Factoring is powerful because it reduces the problem of solving a quadratic equation to finding the zeros of these simpler linear factors. The principle is based on the Zero Product Property, which states that if a product equals zero, then at least one of the factors must be zero. Thus, setting each factor equal to zero and solving for x gives us the roots of the original quadratic equation.

Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a foolproof method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\).

It is derived from the process of completing the square and gives you the roots directly. The formula contains the symbol \(\pm\), which indicates that there are usually two solutions to a quadratic equation: one for when you add the square root and one for when you subtract it.

Advantages of the Quadratic Formula

• It works for all quadratic equations, even when factoring is difficult or impossible.
• It provides an exact solution, assuming the square root can be simplified.
• It is particularly helpful when the coefficients a, b, or c are large numbers or when they do not factor neatly.

The quadratic formula is a reliable tool that students can use to find the roots of a quadratic equation when other methods are not suitable or more challenging to apply.

Simplifying Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction, and they often occur in the solutions to quadratic equations, usually under the square root sign in the quadratic formula. Simplifying irrational numbers involves expressing them in the simplest radical form possible.

Some steps for simplifying irrational numbers include:

• Finding the prime factors of the number under the square root.
• Identifying pairs of prime factors, as each pair can be taken out of the square root as a single number.
• Rewriting the square root with simplified terms outside the radical and any remaining primes inside.

Importance of Simplification

Simplification is crucial as it leads to a more understandable form, which is particularly helpful in further calculations or when graphing the solutions of quadratic equations on a number line or coordinate plane. Moreover, simplified forms are the expected standard in mathematics, ensuring consistency in how solutions are presented.