Problem 75 - Algebra 2

Question

The equation \(x^{2}-3 x+a=0\) has two roots. One root of the equation is \(2 .\) What is the other root? $$ \begin{array}{lllll}{\text { A. }-2} & {\text { B. }-1} & {\text { C. } 1} & {\text { D. } 3}\end{array} $$

Step-by-Step Solution

Answer

The other root of the equation is 1.

Step 1: Use the given root to find the value of a

Substitute x with the given root, 2, into the equation to solve for a. The equation becomes: \(2^{2} - 3(2) + a = 0\), which simplifies to \(4 - 6 + a = 0\). Solving for a gives us \(a = 2\).

Step 2: Use the value of 'a' to solve for the other root

Since the sum of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(-b/a\), we can use this to find the other root. The sum of the roots here is \(-(-3)/1 = 3\). Let the other root be r. So, we have \(2 + r = 3\).

Step 3: Find the other root

Subtracting 2 from both sides of the equation \(2 + r = 3\) gives us the other root \(r = 3 - 2\), which simplifies to \(r = 1\).

Key Concepts

Solving Quadratic Equations

When faced with a quadratic equation, one might feel intimidated, but fear not! These equations are simply polynomials with a degree of two, represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving them involves finding the values of \(x\) that make the equation true, which are called 'roots' or 'solutions'.
To solve such equations, we can utilize various methods like factoring, completing the square, graphing or using the notable quadratic formula. In textbook exercises, the method may be suggested, but understanding multiple approaches enhances your mathematical toolset.

Sum of Roots in Quadratics

A beautiful aspect of quadratic equations is the relationship between their coefficients and their roots. For the standard form \(ax^2 + bx + c = 0\), the sum of the roots, denoted \(\alpha + \beta\), equals \(-\frac{b}{a}\).
This elegant result comes from Vieta's formulas, which are a set of equations relating the coefficients of a polynomial to sums and products of its roots. Knowing this relationship is incredibly useful; it allows us to find one root when the other is known, or even to deduce properties of the roots without directly solving the equation.

Quadratic Formula

One surefire method that solves any quadratic equation is the famed quadratic formula. Given \(ax^2 + bx + c = 0\), the roots are given by: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \].
The formula is derived from completing the square and provides both roots directly. The term under the square root, \(b^2 - 4ac\), known as the discriminant, tells us the nature of the roots. If it's positive, we get two real distinct roots; if zero, one real repeated root; and if negative, two complex roots. This formula is a powerful tool in the algebraic arsenal, treasured for its clear, step-by-step path to the solution.

Algebraic Expressions

At the heart of these quadratic equations are algebraic expressions, the building blocks of algebra. They are composed of variables, coefficients, and arithmetic operations. In the context of quadratic equations, expressions take on the form of \(ax^2 + bx + c\), where each part has a role in molding the graph and the nature of the solutions.
Understanding how to manipulate these expressions is key to solving and graphing quadratic equations. They're also the canvases on which properties like the sum of roots are painted, showcasing the interconnected nature of algebraic concepts.