Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation by factoring. $$ 2 x^{2}=19 x+33 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x= -3/2\) and \(x=11\).

Step 1: Rearrange the Equation

Firstly, rearrange the equation to make one side equal to zero. The equation becomes \(2x^{2} - 19x - 33 = 0\)

Step 2: Factor the Quadratic

Now, factor the quadratic equation. In this case, the factored form of the equation is \( (2x+3)(x-11) = 0\)

Step 3: Solve for x

Having factored the quadratic equation, next is to find the roots by setting each factor equal to zero and solving for \(x\). From \(2x+3=0\) we have \(x= -3/2\), and from \(x-11=0\) we have \(x=11\).

Key Concepts

Factoring Quadratics

Quadratics are polynomial equations of degree 2, typically in the form of \( ax^2 + bx + c = 0 \). Factoring them is one of the methods to solve these equations, which involves expressing the quadratic as a product of two binomials. The quadratic equation provided, \( 2x^2 - 19x - 33 = 0 \), is a perfect candidate for factoring. The key idea is to express this equation in the form \( (px + q)(rx + s) = 0 \). To do this:

• First, multiply the leading coefficient \(2\) by the constant term \(-33\) to get \(-66\).
• Now, find two numbers that multiply to \(-66\) and add to \(-19\): In this case, the numbers are \(3\) and \(-22\).
• Replace \(-19x\) with \(3x - 22x\) to break the middle term, making the equation \(2x^2 + 3x - 22x - 33 = 0\).
• Group terms \((2x^2 + 3x)\) and \((-22x - 33)\), and factor the groups: \( x(2x + 3) - 11(2x + 3) = 0\).
• Now factor out the common binomial to get \((2x + 3)(x - 11) = 0\).

This process of breaking apart the middle term is often known as factoring by grouping, and it's a powerful technique for solving quadratic equations.

Solving Equations

After factoring a quadratic equation, solving it becomes simple. You use the factored form to find the values of \(x\) that make the equation equal to zero. In our example \((2x + 3)(x - 11) = 0\), the solution involves:

• Setting each factor equal to zero, leading to two separate equations: \(2x + 3 = 0\) and \(x - 11 = 0\).
• Solving these straightforward equations will give you the roots. For \(2x + 3 = 0\), rearrange to find \(x = -\frac{3}{2}\). Similarly, for \(x - 11 = 0\), simply add \(11\) to both sides to get \(x = 11\).

These steps isolate \(x\) and determine the values (or solutions) of the quadratic equation. The solution to a quadratic is often two distinct real numbers, a repeated real number, or complex numbers, depending on the equation.

Algebraic Methods

Quadratic equations can be solved through several algebraic methods, each suited to different types of equations or preferences for calculation. Factoring is just one of these methods.Other common algebraic methods include:

• **Completing the Square**: Useful for converting any quadratic equation into a form that can easily reveal the roots, particularly when the equation does not factor neatly.
• **Quadratic Formula**: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). This formula can solve any quadratic equation and is derived from the process of completing the square. It's ideal when factoring is complex or impractical.
• **Graphical Solutions**: Plotting the quadratic function to find where it intersects the x-axis, providing a visual representation of the roots.

Each method has its own advantages and knowing when to apply each can simplify the problem-solving process. Familiarity with multiple strategies enables flexible and efficient problem-solving in algebra.