Problem 64 - Algebra and Trigonometry

Question

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 4 x^{2}-16 x+19=0 $$

Step-by-Step Solution

Answer

The solutions are \(2 + \frac{\sqrt{3}}{2}i\) and \(2 - \frac{\sqrt{3}}{2}i\).

Step 1: Identify the Type of Equation

The given equation is a quadratic equation in the form \(ax^2 + bx + c = 0\), with \(a = 4\), \(b = -16\), and \(c = 19\).

Step 2: Apply the Quadratic Formula

To find the solutions, we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 4\), \(b = -16\), and \(c = 19\) into the formula.

Step 3: Calculate the Discriminant

The discriminant \(D\) is calculated as \(b^2 - 4ac\). Compute \((-16)^2 - 4 \cdot 4 \cdot 19 = 256 - 304 = -48\). Since the discriminant is negative, the solutions will be complex numbers.

Step 4: Simplify the Complex Square Root

Since the discriminant is \(-48\), we need to simplify \(\sqrt{-48}\). This becomes \(\sqrt{48}i\), which is further simplified to \(4\sqrt{3}i\).

Step 5: Compute the Solutions

Substitute into the quadratic formula: \[x = \frac{-(-16) \pm 4\sqrt{3}i}{8}\]. Simplify to find \(x = \frac{16 \pm 4\sqrt{3}i}{8}\), which gives \(x = 2 \pm \frac{\sqrt{3}}{2}i\).

Step 6: Express in Required Form

The solutions are \(x = 2 + \frac{\sqrt{3}}{2}i\) and \(x = 2 - \frac{\sqrt{3}}{2}i\), expressed in the form \(a + bi\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). These equations curve into a parabola when graphed. The quadratic formula allows us to find the solutions (roots) of any quadratic equation. It is expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, \(a\), \(b\), and \(c\) are coefficients from the equation. The symbol \(\pm\) indicates that there are typically two solutions, resulting in two roots for the equation. By substituting the values of \(a\), \(b\), and \(c\) into this formula, we can determine the values of \(x\) that solve the equation. This method works universally, no matter whether the solutions are real or complex numbers.

Complex Numbers

Complex numbers provide a way to handle solutions that aren't real numbers. They are composed of a real part and an imaginary part, expressed in the form \(a + bi\). Here, \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\).

• Real numbers are the set of numbers we are most familiar with, like the numbers on a number line.
• Imaginary numbers are used when solving equations with negative square roots.

In equations like our quadratic equation above, complex numbers arise when the discriminant is negative, indicating that no real roots exist. Thus, understanding complex numbers is crucial in dealing with certain equations.

Discriminant

The discriminant of a quadratic equation is a key component that appears under the square root sign in the quadratic formula. It is calculated as \(b^2 - 4ac\). The discriminant helps determine the nature of the roots of a quadratic equation.

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, there is exactly one real solution (also called a repeated root).
• If the discriminant is negative, the solutions are complex and appear as conjugate pairs.

In our specific problem, the discriminant is \(-48\), indicating that the roots are complex numbers. Recognizing the discriminant's role saves time since it predicts the nature of the solutions even before calculating them fully.

Complex Square Root

Finding the square root of a negative number involves embracing complex numbers. A negative discriminant, as in our example's \(-48\), requires us to compute the complex square root. This is done by factoring out the \(i\), the imaginary unit, since \(\sqrt{-1} = i\). For \(\sqrt{-48}\), one can simplify this to \(\sqrt{48}i\). Breaking it down further, \(\sqrt{48}\) can be expressed as \(4\sqrt{3}\). Hence, the square root of \(-48\) is \(4\sqrt{3}i\). This process of simplification makes it easier when substituting back into the quadratic formula, ensuring we express the roots correctly in the \(a + bi\) form. Understanding how to handle and simplify complex square roots aids in achieving the correct complex solutions.