Problem 43 - Algebra 1: Concepts and Skills

Question

Solve the equation by factoring. $$ 7 x^{2}-10 x+3=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 3/7\) and \(x = 1\).

Step 1: Simplify the equation

The equation is already simplified which is \(7x^{2}-10x+3=0\).

Step 2: Factoring

To solve by factoring, the equation needs to be arranged into two binomials, namely as follows: \((7x - 3)(x - 1)= 0\). The check for this factorization is done by expanding \( (7x - 3)(x - 1) \) which should give back the original quadratic equation.

Step 3: Solving for x

Setting each factor equal to zero gives the solutions of the equation, \[7x - 3 = 0 => x = 3/7\] and \[x - 1 = 0 => x = 1\].

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that express the equality of two expressions with variables involved. The equation presented in our exercise, \(7x^{2} - 10x + 3 = 0\), is specifically a quadratic equation, which generally takes the form \(ax^{2} + bx + c = 0\). Quadratics are special because they graph as parabolas and have unique properties and methods for finding their solutions.
A fundamental goal in solving algebraic equations is to determine the value of the variable that makes the equation true. In our example, the variable \(x\) needs to be determined such that the entire expression equals zero. Working with algebraic equations, it's important to recognize different types of equations, such as linear, quadratic, and polynomial, as this affects the methods you'll use to solve them. Factoring, which we'll discuss in detail, is a key method used for solving them.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions for \(x\) that make the equation true. For the equation \(7x^{2} - 10x + 3 = 0\), the roots can be found by factoring. These roots are also where the parabola, represented by the quadratic equation, crosses the x-axis.
In our quadratic equation, factoring resulted in the expression

• \((7x - 3) = 0\)
• \((x - 1) = 0\)

Solving these simple equations gives us the roots \(x = \frac{3}{7}\) and \(x = 1\). These solutions tell us the specific x-values where the graph of the parabola intersects the x-axis. Understanding how to find these roots not only helps in solving algebraic equations but also aids in graphing and analyzing the behavior of quadratic functions.

Polynomial Factoring

Polynomial factoring involves breaking down a polynomial into simpler "factor" polynomials that multiply together to produce the original polynomial. It's like taking a larger shape apart into smaller blocks. In the context of a quadratic equation, such as \(7x^{2} - 10x + 3\), the goal is to express it as the product of two binomials.
This factoring method leverages the zero-product property, which states that if a product of multiple factors equals zero, at least one of the factors must equal zero. By writing our quadratic as \((7x - 3)(x - 1) = 0\), we effectively take advantage of this property, allowing us to set each binomial to zero separately and solve for \(x\).
The ability to factor is a powerful tool in algebra that simplifies complex expressions and reveals critical information about the solutions to the polynomial equations, such as the roots.