Problem 2 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$x^{2}-8 x-4=0$$

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

In the given quadratic equation, \(x^2 - 8x - 4 = 0\), identify the coefficients: \(a = 1\), \(b = -8\), and \(c = -4\).

Step 2: Choose the quadratic formula method

Since factoring doesn't seem straightforward given these coefficients, let's use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are the coefficients identified.

Step 3: Compute the discriminant

Calculate the discriminant \(b^2 - 4ac\): \[ (-8)^2 - 4(1)(-4) = 64 + 16 = 80 \]

Step 4: Apply the quadratic formula

Substitute the coefficients and discriminant back into the quadratic formula: \[ x = \frac{-(-8) \pm \sqrt{80}}{2(1)} = \frac{8 \pm \sqrt{80}}{2} \]

Step 5: Simplify the expression

Simplify \(\sqrt{80}\) to \(4\sqrt{5}\), which results in: \[ x = \frac{8 \pm 4\sqrt{5}}{2} \] Further simplify: \[ x = 4 \pm 2\sqrt{5} \]

Step 6: State the solutions

The solutions to the quadratic equation \(x^2 - 8x - 4 = 0\) are \(x = 4 + 2\sqrt{5}\) and \(x = 4 - 2\sqrt{5}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that helps to solve any quadratic equation. A quadratic equation generally follows the format: \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are coefficients. When direct factoring or other methods aren't suitable, the quadratic formula comes into play. The formula itself is:

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

It provides the solution to the quadratic equation by giving values of \(x\) that satisfy it. You will see two solutions, thanks to the "\( \pm \)" sign, which indicates we might have different results based on adding or subtracting the square root of the discriminant. This formula is incredibly reliable and is pivotal in solving quadratic equations when other simpler methods like factoring don’t work out.
One of its strengths is that it gives solutions as real or complex numbers, so it covers all bases.

Discriminant

Understanding the discriminant is a key part of solving quadratic equations with the quadratic formula. The discriminant is given by the expression \( b^2 - 4ac \). It is part of the square root in the formula, and it tells us a lot about the nature of the equation's roots. Here’s how:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real double root, meaning both solutions are the same.
• If \( b^2 - 4ac < 0 \), the roots are complex and not real numbers.

In our given solution, the discriminant was calculated to be 80. Since 80 is greater than zero, our equation has two distinct real solutions. This information helps to anticipate the kind of answers you will end up with. Therefore, calculating the discriminant is a crucial step in understanding the set of solutions for any quadratic equation you are working on.

Coefficients

Coefficients are the numbers in front of the variables in polynomial expressions and equations. In a quadratic equation like \( ax^2 + bx + c = 0 \), the coefficients are \(a\), \(b\), and \(c\). These numbers largely determine the form and solution of the equation. Here's a breakdown of their significance:

• \(a\) is the coefficient of \(x^2\), which can modify the parabola's direction and width.
• \(b\) is the coefficient of \(x\), influencing the axis of symmetry of the parabola represented by the equation.
• \(c\) is the constant term, shifting the position of the parabola up or down without affecting its shape.

In our example, \(a = 1\), \(b = -8\), and \(c = -4\). Understanding these coefficients not only helps in employing the quadratic formula accurately but also enriches your grasp of how quadratic equations behave overall. They are fundamental in setting up the equation before solving it using methods like the quadratic formula.