Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(11 x^{2}+33 x=0\)

Step-by-Step Solution

Answer

The solution to the quadratic equation \(2x^{2} + 4x - 9 = 2(x-1)^{2}\) is \(x = \frac{11}{8}\)

Step 1: Simplify the equation

Expand the right-hand side of the equation, so that both sides of the equation are in the same form. So the equation becomes: \(2x^{2} + 4x - 9 = 2(x^{2} - 2x + 1)\). Further simplification will give us \(2x^{2} + 4x - 9 = 2x^{2} - 4x + 2\).

Step 2: Simplify to standard quadratic form

Now that both sides of the equation are in the same form, we can subtract the right-hand side from the left-hand side to get the standard form of quadratic equation. This results in \(8x - 11 = 0\)

Step 3: Solve for x

To solve for x, we simply need to isolate x. To do this, add 11 to both sides of the equation and then divide through by 8. So, \(x = \frac{11}{8}\)

Key Concepts

Standard Quadratic Form

The standard quadratic form is a way of expressing quadratic equations in a specific layout. A quadratic equation in standard form looks like this: \[ ax^2 + bx + c = 0 \]where:

• \( a, b, \text{ and } c \) are constants
• \( x \) represents the unknown variable
• \( a \) should not be zero, because if it is, the equation is no longer quadratic

The coefficients \( a, b, \text{ and } c \) give us much information about the curve's orientation and position, called a parabola, formed when the equation is graphed. The equation entered the standard form in the solution, starting from \( 2x^2 + 4x - 9 \) and simplifying both sides until it resembled \( 0 = ax^2 + bx + c \). Understanding this form is crucial because it sets the stage for solving the equation efficiently.

Solving Quadratic Equations

Solving quadratic equations can be achieved through several methods like factoring, completing the square, or using the quadratic formula. In this case, transforming the original equation into a simpler one allowed direct solving. The equation was initially:\[ 2x^2 + 4x - 9 = 2(x - 1)^2 \]After expansion and simplification, it turned into:\[ 2x^2 + 4x - 9 = 2x^2 - 4x + 2 \]Subtracting the right side from the left, the equation simplified to a linear form:\[ 8x - 11 = 0 \]This step's goal was to isolate \( x \) by performing operations to both sides that maintained equality until \( x \) was by itself on one side of the equation. Since most quadratic equations aren't as lucky as this one to simplify directly to a linear form, having knowledge of other solving methods is essential.

Simplifying Equations

Simplifying equations often involves making them more straightforward, which can reveal the values of unknowns more easily. Simplification might mean expanding expressions, combining like terms, or following logical algebraic steps.In the exercise, the equation:\[ 2(x - 1)^2 = 2x^2 - 4x + 2 \],required expansion to ensure every part of the equation was in terms of single powers of \( x \). This turned the problem into:\[ 2x^2 + 4x - 9 = 2x^2 - 4x + 2 \].Next, subtracting identical terms on both sides was crucial, leading to simpler expressions like the linear equation \( 8x - 11 = 0 \). Strategies like these keep equations manageable and pave the way for solving them. In many mathematical scenarios, the ability to simplify effectively is just as crucial as solving the equation.