Problem 63 - Algebra 2

Question

Solve each equation by completing the square. $$ x^{2}+8 x+20=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -4 + 2i\) and \(x = -4 - 2i\).

Step 1: Move Constant to the Other Side

Start by moving the constant term to the right-hand side of the equation. This makes it easier to complete the square.The given equation is:\[ x^{2} + 8x + 20 = 0 \]Subtract 20 from both sides:\[ x^{2} + 8x = -20 \]

Step 2: Find the Number to Complete the Square

Take the coefficient of \(x\), divide it by 2, and square the result. This is the number we'll add to both sides to complete the square.The coefficient of \(x\) is 8. Divide by 2 and square it:\[ \left( \frac{8}{2} \right)^2 = 4^2 = 16 \]

Step 3: Add the Square to Both Sides

Add the number found in Step 2 to both sides of the equation.Our equation becomes:\[ x^{2} + 8x + 16 = -20 + 16 \]which simplifies to:\[ x^{2} + 8x + 16 = -4 \]

Step 4: Write the Perfect Square Trinomial

Rewrite the left-hand side as a perfect square trinomial.Use the number from Step 2 to form the square:\[ (x+4)^2 = -4 \]

Step 5: Solve for x

Now solve for \(x\) by taking the square root of both sides.\[ \sqrt{(x+4)^2} = \pm\sqrt{-4} \]This implies:\[ x + 4 = \pm 2i \] (since the square root of \(-4\) is \(2i\))

Step 6: Isolate x

Finally, subtract 4 from both sides to solve for \(x\).\[ x = -4 + 2i \] and \[ x = -4 - 2i \]

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra. They are polynomial equations of the form \(ax^2 + bx + c = 0\) with \(a\), \(b\), and \(c\) as constants and \(a eq 0\). Solving quadratic equations can be challenging, but there are various methods to find the roots of these equations.

One popular method is "Completing the Square". This technique transforms the equation into a perfect square trinomial, making it easier to solve. Other methods include factoring (if possible) and using the quadratic formula.

Quadratic equations have different types of solutions based on the discriminant \(b^2 - 4ac\):

• If \(b^2 - 4ac > 0\), there are two real and distinct solutions.
• If \(b^2 - 4ac = 0\), there is one real solution (a repeated root).
• If \(b^2 - 4ac < 0\), the solutions are complex or imaginary numbers.

Understanding these concepts is crucial for solving quadratic equations efficiently.

Complex Numbers

Complex numbers extend the idea of real numbers and are essential when solving equations with no real solutions. A complex number is of the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part; \(i\) is the imaginary unit, defined as \(i^2 = -1\).

In our solution example, the equation \((x+4)^2 = -4\) results in no real numbers since the square of a real number can't be negative. Taking the square root of both sides, we encounter \(\pm \sqrt{-4}\), which simplifies to \(\pm 2i\). This gives the solutions \(x = -4 + 2i\) and \(x = -4 - 2i\).

Complex numbers are graphically represented in a plane, often called the complex plane, with the real and imaginary parts as coordinates. Understanding how to work with complex numbers is crucial for advanced algebra and calculus.

Perfect Square Trinomial

A perfect square trinomial is an essential concept when completing the square. It refers to a quadratic expression that can be factored into a form \((a+b)^2\). For example, \(x^2 + 8x + 16\) is a perfect square trinomial that can be rewritten as \((x+4)^2\).

Creating a perfect square trinomial involves adjusting the quadratic equation by adding and subtracting the same value. In the original equation, \(x^2 + 8x + 20 = 0\), by moving the constant to one side and solving for the right number to add and subtract (in this case \(16\)), we formed the perfect square trinomial. This helps isolate the square and solve the quadratic equation efficiently.

This is a pivotal step in many algebraic problem-solving strategies, ensuring an equation can be easily solved by straightforward operations like extracting square roots.

Square Roots

Square roots are fundamental in algebra and solving quadratic equations. The square root of a number \(a\) is any number \(x\) such that \(x^2 = a\). Not all numbers have real square roots, especially when \(a\) is negative, which leads to imaginary numbers.

In our solution, we encountered \(\sqrt{-4}\), which doesn't have a real square root. Instead, we express it as \(2i\) or \(-2i\), demonstrating the transition from real to complex solutions.

When solving quadratic equations by completing the square, taking the square root is a key step to find the values of \(x\). Often, you will end up with expressions involving \(\pm\) the square root, indicating two potential solutions. Understanding square roots, both real and complex, is crucial for navigating algebra problems.