Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ -x^{2}+8 x=12 $$

Step-by-Step Solution

Answer

The roots of the quadratic equation are \( x = 6 \) and \( x = 2 \).

Step 1: Re-arrange the Equation

Rearrange the given equation to have 0 on one side. Thus, \( -x^2 + 8x - 12 = 0 \)

Step 2: Factorize the Equation

To ease the process of factorization, multiply the entire equation by -1 to get rid of the negative sign before \( x^2 \). This will give us \( x^2 - 8x + 12 = 0 \). Now, factorize this equation into \((x - a)(x - b) = 0\), where a and b are factors of 12 that sums up to 8. The factored form of the equation is \( (x - 6)(x - 2) = 0 \).

Step 3: Find the Roots

The roots of the equation can be found by setting each factor to zero and solving for x. This will give us \( x - 6 = 0 \) and \( x - 2 = 0 \). Solving these we get \( x = 6 \) and \( x = 2 \).

Key Concepts

Quadratic Equation

A quadratic equation is an algebraic expression of the second degree, generally represented in the form (ax^2 + bx + c = 0), where a, b, and c are constants and a ≠ 0. The name 'quadratic' originates from 'quad' meaning square, as the variable gets squared (like x^2).

These equations are solved to find the 'roots' or the 'zeroes' of the quadratic function, which are the values of x that satisfy the equation. There are various methods to solve a quadratic equation, such as factoring, using the quadratic formula, completing the square, or graphing. Each method has its own utility and learning all of them can greatly improve your problem-solving abilities in algebra.

Factorization

The process of factorization, or factoring, is breaking down a complex equation or expression into a product of simpler factors which, when multiplied together, give back the original expression. In the case of quadratic equations, factoring is a valuable skill. It involves finding two binomials that when multiplied give the original quadratic equation. The standard form of factoring a quadratic is (x - p)(x - q) = 0, where p and q are the specific numbers that satisfy the condition that they add up to b and multiply to c, considering our quadratic is in the form x^2 + bx + c.

When the quadratic is set equal to zero, factoring is particularly useful because it allows us to apply the Zero Product Property. According to this property, if the product of two expressions is zero, at least one of the expressions must be zero. This leads directly to the roots of the equation.

Finding Roots of Polynomial

Finding roots of a polynomial refers to determining the values for which the polynomial equals zero. For a quadratic equation, these roots can be understood as the points where the parabola, the graph of the quadratic function, intersects the x-axis.

There are different methods to find these roots. After factoring the quadratic equation, as in the method demonstrated, we use the Zero Product Property to set each factor equal to zero. The solutions to these simple equations are the roots of the original quadratic equation. These roots can be real or complex numbers and a quadratic equation always has two roots, which may or may not be distinct.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and arithmetic operations. In our context, the quadratic equation itself is an algebraic expression involving the square of a variable. Working with algebraic expressions is fundamental in solving polynomial equations like the one we explored.

Understanding how to manipulate these expressions through operations such as expansion, factorization, and simplification is crucial. These skills not only aid in solving equations but also evolve into more advanced topics in mathematics, such as calculus and beyond. Practice and familiarity with algebraic expressions lay the groundwork for these future mathematical endeavors.