Problem 20 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(x^{2}-16 x+4=0\)

Step-by-Step Solution

Answer

The discriminant is 240; the equation has two distinct real roots: \(x = 8 \pm 2\sqrt{15}\).

Step 1: Identify Coefficients

In the quadratic equation \(x^2 - 16x + 4 = 0\), identify the coefficients as \(a = 1\), \(b = -16\), and \(c = 4\).

Step 2: Calculate the Discriminant

Use the formula for the discriminant \(b^2 - 4ac\). Substitute the given coefficients: \((-16)^2 - 4(1)(4) = 256 - 16 = 240\). So, the discriminant is \(240\).

Step 3: Describe the Number and Type of Roots

The discriminant is 240, which is greater than zero. This indicates that the equation has two distinct real roots.

Step 4: Use the Quadratic Formula

Apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values: \(x = \frac{-(-16) \pm \sqrt{240}}{2(1)}\).

Step 5: Simplify the Expression

Calculate \(\sqrt{240}\). Note that \(\sqrt{240} = \sqrt{16 \times 15} = 4\sqrt{15}\). Substitute it back into the quadratic formula to get \(x = \frac{16 \pm 4\sqrt{15}}{2}\).

Step 6: Final Simplification

Simplify the fraction: \(x = 8 \pm 2\sqrt{15}\). Therefore, the exact solutions are \(x = 8 + 2\sqrt{15}\) and \(x = 8 - 2\sqrt{15}\).

Key Concepts

Discriminant

When solving quadratic equations like \(x^2 - 16x + 4 = 0\), the discriminant plays a crucial role. The discriminant is derived from the quadratic formula and is expressed as \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.

The value of the discriminant can reveal a lot about the roots of the equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it equals zero, the equation has exactly one real root (a repeated or double root).
• If negative, there are no real roots, meaning the solutions are complex numbers.

In our example \(x^2 - 16x + 4 = 0\), calculating the discriminant: \((-16)^2 - 4(1)(4) = 256 - 16 = 240\).

Since 240 is positive, the equation has two distinct real roots.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to find the roots by substituting the values of \(a\), \(b\), and \(c\) from the equation into it.

For the equation \(x^2 - 16x + 4 = 0\):

• We substitute in \(a = 1\), \(b = -16\), and \(c = 4\).
• With the discriminant already calculated as 240, it becomes \(x = \frac{-(-16) \pm \sqrt{240}}{2(1)}\).
• This simplifies to \(x = \frac{16 \pm \sqrt{240}}{2}\).

To further simplify, find \(\sqrt{240}\), which is \(\sqrt{16 \times 15} = 4\sqrt{15}\). Plug this back into the equation to get

\(x = \frac{16 \pm 4\sqrt{15}}{2}\), leading to the solutions \(x = 8 + 2\sqrt{15}\) and \(x = 8 - 2\sqrt{15}\).

Real Roots

Real roots are the solutions of a quadratic equation that lie on the real number line. When the discriminant is positive, you'll encounter two distinct real roots. Conversely, if it equals zero, there is one repeated real root.

In the scenario of our equation \(x^2 - 16x + 4 = 0\), since we calculated a positive discriminant of 240:

• This tells us that the roots are real and distinct.
• The roots found with the quadratic formula were \(x = 8 + 2\sqrt{15}\) and \(x = 8 - 2\sqrt{15}\).

These values are solutions that satisfy the original equation when substituted back, confirming they are indeed real. Understanding the nature of roots is crucial for comprehending how quadratic equations behave and interact with the number line.