Problem 50 - Algebra 2

Question

ACT/SAT If one of the roots of the equation \(x^{2}+k x-12=0\) is \(4,\) what is the value of \(k ?\) \(\mathrm{A}-1\) \(\mathrm{B} 0\) \(\mathrm{C} 1\) \(\mathrm{D} 3\)

Step-by-Step Solution

Answer

The value of \(k\) is \(-1\).

Step 1: Use the Root of the Equation

The equation is given as \(x^2 + kx - 12 = 0\). One root of this equation is \(4\). Since \(x = 4\) is a root, substituting \(x = 4\) into the equation gives \(4^2 + 4k - 12 = 0\).

Step 2: Simplify the Equation

Calculate \(4^2\), which equals \(16\). Substitute it into the equation to get: \[16 + 4k - 12 = 0\]. Simplify this to \[4 + 4k = 0\].

Step 3: Solve for k

Isolate \(k\) by subtracting \(4\) from both sides of the equation: \[4k = -4\]. Then, divide both sides by \(4\) to solve for \(k\): \[k = -1\].

Key Concepts

Roots of Equations

Roots of equations are the values of the variable that make the equation true. To put it simply, they are the solutions to the equation. For quadratic equations like the one in our exercise, there can be up to two roots because the highest power of the variable, here denoted as \(x^2\), is two. The roots are the values that you substitute for \(x\) to make the whole equation equal zero. In our example, we were given one of these roots directly, which was \(4\). By knowing at least one root, we can easily simplify and solve the equation to find unknown constants or even the other root. Understanding roots helps us solve all types of equations by breaking them down into smaller, manageable parts.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. In our example, we dealt with the equation \(x^2 + kx - 12 = 0\), where we needed to determine the value of \(k\). This process started by using the given root, \(4\), to substitute into the equation. When substituting, the equation transforms into \(4^2 + 4k - 12 = 0\). From there, it's about simplifying the expression until we solve for the unknown. Quadratic equations can generally be solved using several methods, such as factoring, completing the square, or using the quadratic formula. In this particular example, using the already known root helped us quickly solve for \(k\), demonstrating how knowing one piece of information like a root can make finding unknowns straightforward.

Algebraic Manipulation

Algebraic manipulation refers to the techniques used to rearrange and simplify equations or expressions to make solving them easier. In our exercise, we used substitution and simplification to solve for \(k\). Starting with the equation \(4^2 + 4k - 12 = 0\), we calculated \(4^2\) to get \(16\), transforming the equation to \(16 + 4k - 12 = 0\). With further simplification, the equation became \(4 + 4k = 0\). These steps involve basic operations like addition, subtraction, multiplication, and division. Each manipulation is a logical step that brings us closer to isolating the unknown variable. By performing these operations correctly, we were able to determine the value of \(k\) as \(-1\). Mastering different algebraic manipulations is essential for solving complex equations easily and accurately. This foundational skill helps in tackling more advanced mathematical problems with confidence.