Problem 47 - College Algebra with Modeling and Visualization

Question

Solve the equation by completing the square. $$ x^{2}+4 x-6=0 $$

Step-by-Step Solution

Answer

\(x = -2 \pm \sqrt{10}\)

Step 1: Move constant to the right side

Start by moving the constant term "+6" to the other side of the equation. Therefore, the equation becomes: \[x^2 + 4x = 6\]

Step 2: Complete the square

To complete the square on the left side, take the coefficient of the linear term (4), divide by 2 and square it. \[ \left( \frac{4}{2} \right)^2 = 4 \]Add and subtract this value on the left side. So, the equation becomes:\[x^2 + 4x + 4 - 4 = 6\]

Step 3: Rewrite as a square

Now rewrite the left side as a perfect square and simplify the right side:\[(x + 2)^2 - 4 = 6\]Add 4 to both sides to isolate the perfect square:\[(x + 2)^2 = 10\]

Step 4: Solve for x

Take the square root of both sides:\[x + 2 = \pm \sqrt{10}\]Subtract 2 from both sides to solve for \(x\):\[x = -2 \pm \sqrt{10}\]

Key Concepts

Completing the Square

Completing the square is a useful technique for solving quadratic equations, especially when they do not factor neatly. This technique transforms a quadratic expression into a perfect square trinomial, making it easier to solve.To complete the square:

• First, ensure that the quadratic term's coefficient is 1. If not, divide the entire equation by this coefficient.
• Next, move the constant term to the right side of the equation.
• To find the number that completes the square, take half of the linear term's coefficient and square it.
• Add and subtract this new number within the equation, effectively not changing the equation's value because you are adding zero.

For example, in the equation \(x^2 + 4x - 6 = 0\), after moving the constant to the right and identifying our linear term coefficient (4), we find our completing square term as \( \left( \frac{4}{2} \right)^2 = 4\). This transforms the expression to a perfect square form \((x+2)^2 = 10\), simplifying further computation.

Algebraic Equations

Algebraic equations involve variables raised to various powers and are solved to find the value(s) of these variables. Quadratic equations, like \(ax^2 + bx + c = 0\), have the highest power of 2.In solving algebraic equations, particularly quadratic ones, we use different methods such as factoring, using the quadratic formula, or completing the square. The choice of method can depend on the specific form of the equation.Key features of quadratic equations:

• The discriminant \(b^2 - 4ac\) dictates the nature of the roots. A positive discriminant indicates two real roots, zero gives one, and negative implies complex roots.
• Quadratic equations are vital in modeling scenarios where the change happens quadratically, such as in physics and engineering problems.

Understanding algebraic equations lays the foundation for more advanced algebra and provides critical problem-solving techniques.

Solving Quadratics

Solving quadratic equations can be achieved through various techniques, and understanding which to use helps in finding solutions efficiently. Whether the equation is simple or complex, you will usually end up finding values for \(x\) that satisfy the equation.Common Methods:

• Factoring, useful when the equation can be easily expressed as a product of binomials.
• Quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), works for any quadratic equation, providing direct solutions.
• Completing the Square, as demonstrated earlier, transforms the equation into a form that allows easy isolation of \(x\).

In our exercise of \(x^2 + 4x - 6 = 0\), completing the square resulted in \(x = -2 \pm \sqrt{10}\), providing the solutions in a simplified form. Mastery over these methods enables tackling a variety of quadratic problems with confidence.