Problem 56 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ 3 x^{2}+12 x=-14 $$

Step-by-Step Solution

Answer

No real solution; complex solutions: \(x = -2 \pm i \sqrt{\frac{2}{3}}\).

Step 1: Divide by the Coefficient of the Quadratic Term

First, we need to divide every term in the equation by 3, the coefficient of the quadratic term, to simplify the equation:\[\frac{3x^2}{3} + \frac{12x}{3} = \frac{-14}{3}\]This simplifies to:\[x^2 + 4x = -\frac{14}{3}\]

Step 2: Move the Constant Term to the Other Side

To complete the square, we need to set up the quadratic and linear terms on one side of the equation. Move the constant term to the right side:\[x^2 + 4x + \_ = -\frac{14}{3} + \_\]

Step 3: Complete the Square

To complete the square, take half of the coefficient of the linear term (which is 4), square it, and add it to both sides of the equation.\[\left(\frac{4}{2}\right)^2 = 4\]Add 4 to both sides:\[x^2 + 4x + 4 = -\frac{14}{3} + 4\]

Step 4: Simplify the Right Side

Calculate the right side:\[-\frac{14}{3} + 4 = \frac{-14 + 12}{3} = \frac{-2}{3}\]

Step 5: Write Left Side as Perfect Square

The left side can now be written as a perfect square:\[(x + 2)^2 = \frac{-2}{3}\]

Step 6: Solve for x

Take the square root of both sides to solve for \(x\):\[sqrt{(x+2)^2} = \pm \sqrt{\frac{-2}{3}} \]This implies the square of any real number is non-negative, thus no real solutions. Hence, express in terms of imaginary numbers:\[x + 2 = \pm \sqrt{\frac{-2}{3}} = \pm i \sqrt{\frac{2}{3}}\]Solving for \(x\):\[x = -2 \pm i \sqrt{\frac{2}{3}}\]

Key Concepts

Quadratic Equations

Quadratic equations are one of the fundamental aspects of algebra, often appearing in the form:\[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The standard form makes it easier to apply various algebraic methods to solve the equation, like factoring, using the quadratic formula, or completing the square. The core idea behind a quadratic equation is that the highest exponent of the variable \(x\) is 2.When solving a quadratic equation by completing the square, you first transform it into a format that allows the left side to be expressed as a square. First, divide the entire equation by the leading coefficient (if it is not 1), to simplify the solving process. Next, ensure that all terms with \(x\) are on one side and the constant on the other. This prepares the equation for completing the square, a pivotal step for not just simplifying but also identifying key properties of the quadratic, such as vertex form.Completing the square involves:

• Taking half of the \(b\) coefficient, squaring it, and adding it to both sides
• Writing the left side as a squared binomial
• Solving for \(x\) by taking the square root and simplifying further

This process illustrates why quadratic equations are so powerful in algebra, providing real and complex solutions.

Imaginary Numbers

Imaginary numbers often enter the scene when solving quadratic equations results in taking the square root of a negative number. Traditionally, the set of real numbers doesn't account for this, but imaginary numbers offer a logical extension to explore "non-real" solutions.The notion of an imaginary number is represented by \(i\), where \(i = \sqrt{-1}\). This allows us to express the square root of any negative number as a multiple of \(i\). For instance:

• \(\sqrt{-4} = 2i\)
• \(\sqrt{-9} = 3i\)

In the quadratic equation we're examining, taking the square root of \(\frac{-2}{3}\) results in \(i\sqrt{\frac{2}{3}}\). This shows that our equation lacks any real solutions but instead holds complex solutions involving imaginary numbers.These complex numbers are expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary component. Solving quadratics in this way reveals the complex plane, expanding our toolset for interpreting mathematical problems.

Algebraic Solutions

Algebraic solutions hinge on systematically applying algebraic rules to isolate variables and solve equations. In the context of quadratic equations, techniques like factoring, the quadratic formula, and completing the square are all utilized to find the values of \(x\) that satisfy the equation.For our exercise, completing the square was the chosen method. It involved creating a perfect square trinomial on the left-hand side of the equation:

• First, simplify the equation by ensuring the coefficient of \(x^2\) is 1
• Then, move the constant term to the other side of the equation
• Add and subtract the necessary term to complete the square
• Rewrite the equation such that the left-hand side becomes a square of a binomial

This method not only solves the problem but also enriches one's understanding of quadratic equations as a whole.By the final step, after solving \((x + 2)^2 = \frac{-2}{3}\), we found solutions involving imaginary numbers, indicating a complex solution. This highlights that algebraic solutions aren't restricted to real numbers, but include a broader spectrum, ensuring every quadratic equation can indeed be solved.