Problem 62 - Calculus: One and Several Variables

Question

Find the real roots of the equation. \(x^{2}+8 x+16=0\).

Step-by-Step Solution

Answer

The real root of the equation \(x^{2} + 8x + 16 = 0\) is \(x = -4\).

Step 1: Identify the coefficients of the given equation

The given quadratic equation is in standard form \(ax^{2}+bx+c=0, where \(a=1\), \(b=8\), and \(c=16\) which are the coefficients.

Step 2: Use the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). Plugging in the given values we have \(x = \frac{-8 \pm \sqrt{(8^{2} - 4*1*16)}}{2*1}\).

Step 3: Evaluate the expression under the square root

Evaluate \(b^{2} - 4ac\) where \(b=8\), \(a=1\), and \(c=16\). This gives, \(64 - 64 = 0\).

Step 4: Substitute the evaluated value under the square root into the quadratic formula and solve for x

Using the calculated value, the formula becomes \(x = \frac{-8 \pm \sqrt{0}}{2}\), which simplifies to \(x = -4\) since \(\sqrt{0}=0\).

Key Concepts

Real Roots

In algebra, **real roots** refer to solutions of an equation that are real numbers. These roots appear in quadratic equations when you create a balance in the equation and solve for unknown variables. Real roots are crucial because they represent where a parabola crosses the x-axis on a coordinate plane.

For the equation \(x^2 + 8x + 16 = 0\), the real root calculated is \(-4\). This happens because

• The expression under the square root (the discriminant) in the quadratic formula, \(b^2 - 4ac\), equates to zero.
• If the discriminant is zero, the quadratic equation has exactly one real root, because the plus/minus term in the formula eliminates the need for multiple solutions.

This single real root or overlapping root (also called a double root), \(-4\), indicates that the vertex of the parabola is tangent to the x-axis at that point, effectively touching it at one unique spot.

Quadratic Formula

The **quadratic formula** is a must-know tool for solving quadratic equations. It is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula allows you to find the roots of any quadratic equation, making it incredibly versatile.

For the equation \(x^2 + 8x + 16 = 0\), substituting the values where \(a = 1\), \(b = 8\), and \(c = 16\), leads to the solution:

• \(x = \frac{-8 \pm \sqrt{64 - 64}}{2}\)

Because the discriminant \(64 - 64\) is zero, the formula simplifies to one root, \(x = -4\).

The quadratic formula is especially useful because it works even when factoring is difficult or impossible, providing an exact answer swiftly.

Standard Form of a Quadratic Equation

The **standard form of a quadratic equation** is written as \(ax^2 + bx + c = 0\). This is the general format where:

• \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the variable.
• \(a\) is not zero to ensure the equation remains quadratic.

Understanding the standard form aids in recognizing what the equation represents and how different pieces (coefficients) influence the shape and position of the parabola it describes.

In the exercise, the equation \(x^2 + 8x + 16 = 0\) is already in standard form, making it easier to identify coefficients:

• \(a = 1\)
• \(b = 8\)
• \(c = 16\)

Once identified, you can effectively use the coefficients for further calculations, such as determining the real roots using the quadratic formula or analyzing the parabola's properties, like its vertex and direction.