Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ x^{2}+8 x-4=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -4 + 2\sqrt{5}\) and \(x = -4 - 2\sqrt{5}\).

Step 1: Identify the values of \(a\), \(b\), and \(c\)

From the equation \(x^{2}+8 x-4=0\), it's clear that \(a = 1\), \(b = 8\), and \(c = -4\)

Step 2: Substitute the values into the quadratic formula

Plugging these values into the quadratic formula, we get \[x = \frac{-8 ± \sqrt{(8)^2 - 4*1*(-4)}}{2*1}\]

Step 3: Simplify the equation

Simplify the equation to get \[x = \frac{-8 ± \sqrt{64 + 16}}{2} = \frac{-8 ± \sqrt{80}}{2} = \frac{-8 ± 4\sqrt{5}}{2}\]

Step 4: Solve for \(x\)

Finally, solve for \(x\) to get \[x = -4 ± 2\sqrt{5}\]

Key Concepts

Quadratic Equation

Quadratic equations are polynomial equations of degree two. They take the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. Here, \( a \) cannot be zero since that would make it a linear equation instead. Quadratic equations are noted for their parabolic graph shapes.

To solve quadratic equations, various methods can be deployed:

• Factoring: Breaking down the equation into simpler factors if possible.
• Completing the square: Transforming the equation into a perfect square trinomial.
• Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which can always be used.

In this exercise, we focus on employing the quadratic formula, a reliable method when factoring is complex or not feasible.

Roots of Quadratic Equation

The roots or solutions of a quadratic equation are the values of \( x \) such that the equation holds true. The quadratic formula helps find these roots.

The discriminant, \( b^2 - 4ac \), plays a crucial role in determining the nature of the roots. Depending on its value:

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

For the equation \( x^2 + 8x - 4 = 0 \), the discriminant calculates to \( 64 + 16 = 80 \). A positive discriminant implies two distinct real roots. Simplified further, the equation roots are expressed as \( x = -4 \pm 2\sqrt{5} \).

Algebraic Solution Steps

The algebraic solution steps involve using the quadratic formula efficiently to solve the equation. Let’s delve into the steps with clarity:

1. **Identify \( a \), \( b \), and \( c \) from the equation**:
In \( x^2 + 8x - 4 = 0 \),\( a = 1 \), \( b = 8 \), and \( c = -4 \).
2. **Substitute into the quadratic formula**:
The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) becomes \( x = \frac{-8 \pm \sqrt{64 + 16}}{2} \).
3. **Simplify the discriminant**:
Simplifying, you find \( \sqrt{80} = 4\sqrt{5} \), leading to \( x = \frac{-8 \pm 4\sqrt{5}}{2} \).
4. **Resolve the equation**:
The final simplification yields the roots \( x = -4 \pm 2\sqrt{5} \), thus revealing the values of \( x \) that satisfy the original equation.By following these steps, the equation is solved systematically, providing clarity and ensuring the accuracy of the roots.