Problem 66 - Elementary Algebra

Question

\(5 d^{2}-3 d+2=0\)

Step-by-Step Solution

Answer

The solutions are \( d = \frac{3 + i\sqrt{31}}{10} \) and \( d = \frac{3 - i\sqrt{31}}{10} \).

Step 1: Identify the coefficients

In the quadratic equation, identify the coefficients of the terms. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). Here, the coefficients are: \(a = 5\), \(b = -3\), and \(c = 2\).

Step 2: Use the quadratic formula

The quadratic formula is given by \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \). Substitute the values from Step 1 into this formula: \( d = \frac{{3 \pm \sqrt{{(-3)^2 - 4 \cdot 5 \cdot 2}}}}{2 \cdot 5} \).

Step 3: Simplify under the square root

Simplify the expression under the square root (the discriminant): \( (-3)^2 - 4 \cdot 5 \cdot 2 = 9 - 40 = -31 \). Now the quadratic formula becomes: \( d = \frac{{3 \pm \sqrt{-31}}}{10} \).

Step 4: Express in terms of complex numbers

Since the discriminant is negative, the solutions will be complex numbers. Recall that \( \sqrt{-31} = i\sqrt{31} \) where \(i\) is the imaginary unit. Thus, the solutions are: \( d = \frac{{3 \pm i\sqrt{31}}}{10} \).

Key Concepts

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations. These are equations in the form of \(ax^2 + bx + c = 0\). The quadratic formula can be written as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]. This formula gives you the solutions (or roots) of the quadratic equation.
To use it, you just need to identify the coefficients a, b, and c from your equation.
Then, plug these values into the formula.
Don't forget to take care of the \(\pm\) symbol, which means you will generally have two solutions: one involving addition and one involving subtraction.

complex numbers

Complex numbers involve both real and imaginary parts.
They are written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\).
When the discriminant in the quadratic formula (\(b^2 - 4ac\)) is negative, the solution will involve complex numbers.
For instance, if \(\sqrt{-31} = i\sqrt{31}\), then the solutions are \(d = \frac{{3 \pm i\sqrt{31}}} {10}\).
This means the equation has two complex solutions: \(d = \frac{3 + i\sqrt{31}}{10}\) and \(d = \frac{3 - i\sqrt{31}}{10}\).

discriminant

The discriminant in the quadratic formula is the part under the square root symbol, \(b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation.
If the discriminant is positive (\(b^2 - 4ac > 0\)), you will have two distinct real roots.
If it's zero (\(b^2 - 4ac = 0\)), you will have exactly one real root (or a repeated root).
If it's negative (\(b^2 - 4ac < 0\)), the roots will be complex. For example, in the equation \(5d^2 - 3d + 2 = 0\), the discriminant is \(-31\).
Since this is negative, the roots are complex, \(d = \frac{3 \pm i\sqrt{31}}{10}\).
Understanding the discriminant is crucial for predicting the type of solutions without fully solving the equation.