Problem 52 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 5 x^{2}-7 x+5=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions; it has two complex solutions.

Step 1: Identify the Type of Equation

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \) with the coefficients: \( a = 5 \), \( b = -7 \), and \( c = 5 \).

Step 2: Calculate the Discriminant

To determine the nature of the solutions, calculate the discriminant \( \Delta \) using the formula \( \Delta = b^2 - 4ac \). Here, \( \Delta = (-7)^2 - 4 \times 5 \times 5 = 49 - 100 = -51 \).

Step 3: Analyze the Discriminant

Since the discriminant \( \Delta = -51 \) is less than zero, this indicates that the quadratic equation has no real solutions. Instead, it has two complex conjugate solutions.

Key Concepts

Understanding the Discriminant

The discriminant is a key concept when solving quadratic equations, as it helps us determine the nature of the roots. You can find the discriminant by using the formula \( \Delta = b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), it has exactly one real root, or a repeated root.
• If \( \Delta < 0 \), the equation has no real roots but two complex solutions.

Calculating the discriminant is often the critical first step. It guides whether to proceed with finding real roots or exploring complex solutions.

Complex Solutions Explained

When the discriminant is negative, as in this exercise where \( \Delta = -51 \), the quadratic equation has complex solutions. This means that the roots are not real numbers, and instead exist as complex numbers.
Complex solutions for a quadratic equation are typically in the form \( p \pm qi \), where \( i \) represents the imaginary unit, defined by the property \( i^2 = -1 \).

• The real part \( p \) can be found using \( -\frac{b}{2a} \).
• The imaginary part \( q \) results from \( \frac{\sqrt{|\Delta|}}{2a} \), where \( |\Delta| \) is the absolute value of the discriminant.

The complex solutions are termed conjugates, meaning one will be \( p + qi \) and the other \( p - qi \), which confirms the nature of the symmetry in complex numbers.

Nature of Roots in Quadratic Equations

The nature of the roots of a quadratic equation gives insight into the type and number of solutions based on the discriminant's value. Let's explore what this tells us:

• Two Distinct Real Roots: Occurs when the discriminant is positive. The graph of the quadratic equation will intersect the x-axis at two points.
• One Real Repeated Root: Happens when the discriminant is zero, resulting in the parabola touching the x-axis exactly once at its vertex.
• Two Complex Conjugate Roots: This is the case when the discriminant is negative, as in the given quadratic. The graph of such an equation does not intersect the x-axis since the solutions are not real numbers.

Therefore, understanding the nature of roots is crucial to anticipating the types of numbers—real or complex—you'll be working with in a quadratic equation.