Question

Solve the equation by factoring. $$8 x^{2}-34 x+24=-11$$

Short answer

The solution to the equation are \( x = 5/4 \) and \( x = 7/2 \)

Step-by-step solution

Rearrange the Equation to Standard Form

Subtract the constant on the right side of the equation from both sides to set the equation to zero: \n\( 8x^{2} - 34x + 24 - (-11) = 0 \)\nwhich simplifies to: \n\( 8x^{2} - 34x + 35 = 0 \)

Factor the Quadratic Expression

Factor the quadratic equation. This equation can be factored to: \n\( (4x - 5)(2x - 7) = 0 \)

Use the Zero Product Property

Set each factor in the equation equal to zero and solve for \( x \): \nIf \( 4x - 5 = 0 \), then \( x = 5/4 \)\nIf \( 2x - 7 = 0 \), then \( x = 7/2 \)

Key concepts

Factoring Quadratic Expressions

When you encounter a quadratic equation such as \( 8x^{2} - 34x + 35 = 0 \), the first strategy often involves breaking it down into simpler components through factoring. Factoring quadratic expressions is like decomposing a complex puzzle into smaller, manageable pieces.

To successfully factor a quadratic expression, you should look for two binomials (expressions with two terms) that when multiplied together, give you the original quadratic equation. In the case of our exercise, the equation was factored into \( (4x - 5)(2x - 7) = 0 \). To reach this stage, you can use various methods like the coefficient comparison method, the ac method, or by intelligent guesswork based on factors of the constant term and the middle coefficient's relationship. The goal when factoring is to rewrite the complex equation in a form where you can then apply the zero product property.

Zero Product Property

The zero product property is a fundamental concept in algebra which states that if the product of two factors is equal to zero, then at least one of the factors must be zero. It is essential when solving factored quadratic equations.

Consider the factored form of our quadratic equation: \( (4x - 5)(2x - 7) = 0 \). Using the zero product property, you separate the equation into two possibilities: either \( 4x - 5 = 0 \) or \( 2x - 7 = 0 \). Then, you solve each linear equation for the variable \( x \). This is how the two possible solutions to the original quadratic equation are uncovered. It is a simple yet powerful tool that transforms the problem from a quadratic to two linear problems.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is an essential starting point for solving these types of equations. It is typically represented as \( ax^2 + bx + c = 0 \), where 'a', 'b', and 'c' are constants, and 'x' represents the variable to be solved.

The initial step in our exercise involved reorganizing the equation to get it into this standard form: \( 8x^{2} - 34x + 24 - (-11) = 0 \), which simplified to the standard form of \( 8x^{2} - 34x + 35 = 0 \). This reorganization is crucial as it sets the stage for applying the techniques of factoring. When solving algebraic equations, always aim to start with the standard form to streamline the process of finding the solution.

Algebraic Problem Solving

The art of algebraic problem solving involves a series of logical steps to find the values of unknowns. In quadratic equations, these steps include bringing the equation to standard form, factoring, and then using the zero product property to find solutions.

In our example, once the equation was in standard form and factored, we set each binomial equal to zero to solve for \( x \). The final solutions were \( x = 5/4 \) and \( x = 7/2 \). Remember, the steps must be followed in order, and each should be completed with care to ensure accuracy. Picturing every algebraic problem as a journey from complexity to simplicity can help maintain clarity and focus throughout the solving process.