Problem 38 - Algebra 2

Question

Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary. $$ x^{2}-12 x+36=0 $$

Step-by-Step Solution

Answer

The discriminant is 0, indicating one real solution.

Step 1 Title - Identify the coefficients

The first step is to identify the coefficients of the quadratic equation of the form \( ax^2 + bx + c = 0 \). For the equation \( x^{2}-12x+36=0 \), the coefficients are \( a = 1 \), \( b = -12 \), and \( c = 36 \).

Step 2 Title - Calculate the discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^{2}-4ac \). Substitute the identified coefficients into this formula to find the discriminant. Thus, \( (-12)^{2}-4(1)(36) = 144-144 = 0 \).

Step 3 Title - Determine the number and type of solutions

Since the discriminant is zero, \( \Delta = 0 \), it indicates that the quadratic equation has one real solution (a repeated real root), also known as a 'perfect square trinomial'.

Key Concepts

Quadratic Equation Solutions

Quadratic equations, which take the form of \( ax^2 + bx + c = 0 \), are fundamental in algebra. Solutions to these equations, also termed as 'roots', can be determined using various methods, including factoring, completing the square, the quadratic formula, or graphing.

To predict the number of solutions without solving the equation, we use the discriminant \( \Delta = b^2 - 4ac \). The value of the discriminant provides crucial information:

• If \( \Delta > 0 \), the equation has two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution.
• If \( \Delta < 0 \), the equation has two complex (imaginary) solutions.

In the example \( x^2 - 12x + 36 = 0 \), calculating \( \Delta \) yielded 0, indicating only one real solution exists. This aligns with the fact that the quadratic is a perfect square trinomial.

Real and Imaginary Solutions

The nature of a quadratic equation's solutions is dictated by the discriminant, \( \Delta \). Real solutions correspond to points where the graph of the quadratic equation intersects the x-axis. When \( \Delta > 0 \), the graph cuts the axis at two points, while a zero discriminant \( \Delta = 0 \) suggests a single point of contact, often referred to as a 'double root'.

On the other hand, a negative discriminant \( \Delta < 0 \) indicates that the equation's graph does not touch the x-axis at all. In such situations, the quadratic equation has two imaginary solutions, which, when added or multiplied, reflect certain symmetries. These solutions are complex numbers and are valuable in advanced fields of mathematics and science, despite their 'imaginary' moniker.

Perfect Square Trinomial

A perfect square trinomial is a special form of a quadratic equation, which is created when a binomial is squared. It takes the form \( (ax + b)^2 = ax^2 + 2abx + b^2 \). Predictably, it factors into \( (ax + b)(ax + b) \) and has one unique solution: \( x = -b/a \).

This type of trinomial is recognized by its specific form and by calculating the discriminant, which will be zero. The given example \( x^2 - 12x + 36 = 0 \) represents the squared binomial \( (x - 6)^2 \), confirming it is a perfect square trinomial.

The exercise demonstrates that identifying and understanding the structure of a quadratic equation can significantly simplify the process of finding solutions. It presents an opportunity to practice not only direct solving techniques but also to enhance problem-solving skills by recognizing underlying patterns.