Problem 38 - Algebra for College Students

Question

Use the method of completing the square to solve each quadratic equation. $$2 x^{2}+7 x-3=0$$

Step-by-Step Solution

Answer

\( x = \frac{-7 \pm \sqrt{73}}{4} \).

Step 1: Divide by 2

\( x^2 + \frac{7}{2}x - \frac{3}{2} = 0 \), so \( x^2 + \frac{7}{2}x = \frac{3}{2} \).

Step 2: Complete the square

Half of \( 7/2 \) is \( 7/4 \), squared is \( 49/16 \).
\( x^2 + \frac{7}{2}x + \frac{49}{16} = \frac{3}{2} + \frac{49}{16} = \frac{24+49}{16} = \frac{73}{16} \).

Step 3: Solve

\( \left(x + \frac{7}{4}\right)^2 = \frac{73}{16} \), so \( x = -\frac{7}{4} \pm \frac{\sqrt{73}}{4} = \frac{-7 \pm \sqrt{73}}{4} \).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It typically appears in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The highest exponent of the variable \( x \) is 2, which is what makes it a "quadratic," derived from the Latin word "quadratus" meaning "square." Quadratic equations appear in many real-world scenarios, such as calculating areas, determining the trajectory of an object, or modeling business profits and losses. Understanding how to solve these equations is essential in various fields like physics, engineering, and finance.

Standard Form

The standard form of a quadratic equation is represented as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, with \( a eq 0 \) to ensure the equation is indeed quadratic, not linear. This form is useful for applying different methods of solving equations, such as factoring, completing the square, and using the quadratic formula. Having a quadratic equation in standard form helps identify the coefficients easily, which is crucial for further calculations and understanding the relationship between the coefficients and the equation's graph, a parabola.

Step by Step Solution

Solving a quadratic equation by completing the square is a sequential process that transforms the equation into a perfect square trinomial. Here's a simple step-by-step:

• Ensure the quadratic equation is in standard form. The coefficient of \( x^2 \) should be 1. If it is not, divide the entire equation by \( a \), the coefficient of \( x^2 \).
• Move the constant term \( c \) to the right side of the equation.
• Take half of the coefficient of \( x \), square it, and add this square to both sides of the equation. This completes the square on the left side.
• Rewrite the left side of the equation as a squared binomial and simplify the right side.
• Solve for \( x \) by taking the square root of both sides and then solving the resulting linear equations.

Completing the square is often used when the quadratic formula is not readily applicable or when a more algebraic, methodical approach is desired.

Solve Quadratic Equation

To solve a quadratic equation such as \( 2x^2 + 7x - 3 = 0 \) by completing the square, follow these steps:

• First, the equation is already in standard form, so divide the entire equation by 2, the coefficient of \( x^2 \), to normalize the coefficient to 1: \( x^2 + \frac{7}{2}x - \frac{3}{2} = 0 \).
• Rearrange to isolate the constant: \( x^2 + \frac{7}{2}x = \frac{3}{2} \).
• To complete the square, take half of \( \frac{7}{2} \), which is \( \frac{7}{4} \), square it to get \( \frac{49}{16} \), and add to both sides: \( x^2 + \frac{7}{2}x + \frac{49}{16} = \frac{3}{2} + \frac{49}{16} \).
• Simplify the equation: \( (x + \frac{7}{4})^2 = \frac{73}{16} \).
• Finally, solve for \( x \) by taking the square root: \( x + \frac{7}{4} = \pm \sqrt{\frac{73}{16}} \), leading to the solutions \( x = -\frac{7}{4} \pm \frac{\sqrt{73}}{4} \).

This method serves as an alternative to using the quadratic formula and provides the same reliable results for solving quadratic equations.