Problem 59 - Algebra and Trigonometry

Question

Find all solutions of the equation and express them in the form \(a+b i .\) $$ 2 x^{2}-2 x+1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1}{2} + \frac{1}{2}i \) and \( x = \frac{1}{2} - \frac{1}{2}i \).

Step 1: Identify Coefficients

For the quadratic equation given: \( ax^2 + bx + c = 0 \), identify the coefficients. Here, \( a = 2 \), \( b = -2 \), and \( c = 1 \).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation is given by \( \Delta = b^2 - 4ac \). Substitute the coefficients into the discriminant formula:\( \Delta = (-2)^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 \).

Step 3: Assess the Nature of the Roots

Since the discriminant \( \Delta = -4 \) is negative, the quadratic equation has two complex roots.

Step 4: Use the Quadratic Formula

The solutions for a quadratic equation \( ax^2 + bx + c = 0 \) are given by the quadratic formula:\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \].

Step 5: Insert Values into Quadratic Formula

Substitute \( b = -2 \), \( \Delta = -4 \), and \( a = 2 \) into the quadratic formula:\[ x = \frac{2 \pm \sqrt{-4}}{4} \].

Step 6: Simplify the Solution

Simplify \( \sqrt{-4} \) as \( 2i \), where \( i \) is the imaginary unit. Substitute into the equation:\[ x = \frac{2 \pm 2i}{4} \].This simplifies to two solutions:\[ x = \frac{2}{4} + \frac{2i}{4} = \frac{1}{2} + \frac{i}{2} \]\[ x = \frac{2}{4} - \frac{2i}{4} = \frac{1}{2} - \frac{i}{2} \].

Step 7: Express Solutions in Standard Form

The solutions in the standard form \( a + bi \) are:\[ x = \frac{1}{2} + \frac{1}{2}i \]\[ x = \frac{1}{2} - \frac{1}{2}i \].

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool used to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula can be used whether the roots are real or complex. It is represented as:

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

Here, \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation. The symbol \( \pm \) indicates that there are generally two solutions: one with addition and one with subtraction. By using this formula, you can solve any quadratic equation just by plugging in the values of \( a \), \( b \), and \( c \).
The formula takes into consideration the discriminant \( \Delta = b^2 - 4ac \) to determine the nature of the roots, giving a clear method for finding the values of \( x \) where the function equals zero.

Discriminant

The discriminant is a fundamental concept when working with the quadratic formula, as it helps predict the nature and number of roots for any quadratic equation. Calculated as \( \Delta = b^2 - 4ac \), the discriminant provides the following insights:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( \Delta < 0 \), the equation has two complex roots.

In our problem, the discriminant is \( -4 \), which is less than zero. This indicates that the quadratic equation will have two complex roots. Complex roots always come in conjugate pairs, meaning if one root is \( a + bi \), the other root will be \( a - bi \). This understanding helps us to apply the right strategy when using the quadratic formula, especially in cases involving complex numbers.

Complex Roots

Complex roots emerge from quadratic equations when the discriminant \( \Delta \) is negative. This implies that the roots are not real numbers, but involve the imaginary unit \( i \), where \( i^2 = -1 \). In the context of the quadratic formula, complex roots occur when the quantity under the square root, known as the discriminant, is less than zero.

• This necessitates using the extended formula: \( \sqrt{-n} = i\sqrt{n} \).

For example, in the given exercise, since \( \Delta = -4 \), we rewrite \( \sqrt{-4} \) as \( 2i \). By inserting this into the quadratic formula:

• \( x = \frac{2 \pm 2i}{4} \)

we derive the solutions: \( x = \frac{1}{2} + \frac{1}{2}i \) and \( x = \frac{1}{2} - \frac{1}{2}i \). This forms a pair of complex solutions that appear symmetrically around the real axis on the complex plane. Understanding complex roots enhances our comprehension of how quadratic equations can have deeper mathematical insights beyond just real solutions.