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Chapter 7: Problem 22

In Exercises \(11-30,\) solve the given quadratic equations by completing the square. Exercises \(11-14\) and \(17-20\) may be checked by factoring. $$3 x^{2}=3-4 x$$

Short Answer

Expert verified
The solutions are \(x = \frac{-2 + \sqrt{13}}{3}\) and \(x = \frac{-2 - \sqrt{13}}{3}\).

Step by step solution

01

Rearrange the Equation

First, rearrange the equation \(3x^2 = 3 - 4x\) into the standard quadratic form \(ax^2 + bx + c = 0\). Begin by moving all terms to one side of the equation, resulting in \(3x^2 + 4x - 3 = 0\).
02

Divide by the Coefficient of x虏

Divide all terms by the coefficient of \(x^2\), which is 3, to simplify the equation. \(x^2 + \frac{4}{3}x = 1\).
03

Complete the Square

To complete the square, take half of the coefficient of \(x\) (which is \(\frac{4}{3}\)), square it, and add to both sides. \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\). Add \(\frac{4}{9}\) to both sides: \(x^2 + \frac{4}{3}x + \frac{4}{9} = 1 + \frac{4}{9}\).
04

Simplify and Factor

The left side of the equation now is a perfect square trinomial, so factor it: \((x + \frac{2}{3})^2\). Simplify the right side: \(\frac{9}{9} + \frac{4}{9} = \frac{13}{9}\). Thus, \((x + \frac{2}{3})^2 = \frac{13}{9}\).
05

Solve for x

Take the square root of both sides: \(x + \frac{2}{3} = \pm\sqrt{\frac{13}{9}}\). This gives \(x + \frac{2}{3} = \pm\frac{\sqrt{13}}{3}\). Next, solve for \(x\) by subtracting \(\frac{2}{3}\) from both sides: \(x = -\frac{2}{3} \pm \frac{\sqrt{13}}{3}\). Thus, the solutions are \(x = \frac{-2 + \sqrt{13}}{3}\) and \(x = \frac{-2 - \sqrt{13}}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a technique used to solve quadratic equations, particularly when factoring is difficult or impossible. This method transforms a quadratic equation into a perfect square trinomial.
To complete the square:
  • Start by rearranging the equation into the form: \(ax^2 + bx = c\).
  • Take the coefficient of \(x\), divide it by 2, and then square it. This value is added to both sides of the equation.
  • This process converts the left side into a perfect square trinomial, which can be factored into a binomial squared, such as \((x + rac{b}{2a})^2\).
  • Solve the resulting equation by isolating the variable \(x\).
Completing the square is a handy tool as it also leads naturally into the derivation of the quadratic formula, making it versatile and educational.
Factoring
Factoring is a quick and efficient way to solve quadratic equations, especially when the equation can be easily expressed as a product of binomials. This method relies heavily on identifying two numbers that add up to the coefficient of \(x\) and multiply to the constant term.
To factor a quadratic equation:
  • Ensure that the equation is in standard quadratic form, \(ax^2 + bx + c = 0\).
  • Look for two numbers that multiply to \('ac'\) and add to \(b\).
  • Once found, rewrite the middle term using these two numbers.
  • Factor by grouping or directly factor into two binomials.
Factoring is effective for simple quadratics but becomes less feasible with complex numbers or when no integer factors exist.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Several methods are available, such as factoring, completing the square, and using the quadratic formula.
To solve a quadratic equation:
  • Try factoring first if it's possible, as it provides the quickest solution.
  • If factoring doesn't work, use the completing the square method to reformat the equation into a solvable form.
  • The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), offers a foolproof solution for any quadratic equation.
Knowing when to apply each method enhances problem-solving skills and enables a more efficient solving process.
Standard Quadratic Form
The standard quadratic form is the foundation of all quadratic equations and is written as \(ax^2 + bx + c = 0\), with \(a\), \(b\), and \(c\) being constants where \(a \eq 0\).
  • This form is crucial because it dictates the methods used for solving the equation.
  • It provides a clear structure for identifying the coefficients required for various solving techniques, such as the quadratic formula or completing the square.
  • Rewriting any given quadratic equation into this form is often the first step in the solving process, ensuring consistency and accuracy while applying algebraic methods.
Practicing converting equations to this form enhances algebraic manipulation skills and prepares students to tackle any quadratic problem efficiently.

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