Problem 66 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}=6 x-9 $$

Step-by-Step Solution

Answer

The equation has one real solution.

Step 1: Write the equation in standard form

The given equation is \( x^2 = 6x - 9 \). First, we need to rewrite it in the standard quadratic form, \( ax^2 + bx + c = 0 \). Subtract \( 6x \) and add \( 9 \) to both sides of the equation to get \( x^2 - 6x + 9 = 0 \). Here, \( a = 1 \), \( b = -6 \), and \( c = 9 \).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( b^2 - 4ac \). Substitute \( a = 1 \), \( b = -6 \), and \( c = 9 \) into the discriminant formula: \((-6)^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0 \).

Step 3: Interpret the discriminant

The value of the discriminant is 0. This means that the quadratic equation has exactly one real solution. When the discriminant is zero, the solutions to the equation are real and equal.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). It represents a parabolic curve in a coordinate plane.
Quadratic equations are called 'quadratic' because the highest power of the variable (usually \( x \)) is squared or raised to the power of two.

• Every quadratic equation yields a graph that is a parabola.
• The solutions of a quadratic equation are called "roots" and can be found using various methods including factoring, completing the square, and the quadratic formula.

Understanding these equations is crucial as they frequently appear in different fields such as physics, engineering, and economics.

Exploring Real Solutions Using Discriminant

To determine how many and what type of solutions a quadratic equation has, we can use the discriminant. The discriminant is part of the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), specifically the part under the square root: \( b^2 - 4ac \).

• If the discriminant is positive, the quadratic equation has two distinct real solutions. This occurs because the square root of a positive number is real.
• If the discriminant is zero, there is exactly one real solution. The parabola just touches the x-axis at one point.
• If the discriminant is negative, the equation has no real solutions, as you can't take the square root of a negative number using real numbers. These would be complex or imaginary solutions instead.

In our previous example, with a discriminant of zero, the equation \( x^2 - 6x + 9 = 0 \) has one real, repeated solution.

Writing Equations in Standard Form

A quadratic equation can initially appear in a non-standard form. To analyze or solve it, it's often essential to rewrite it in the standard form \( ax^2 + bx + c = 0 \). This standardization helps apply consistent methods and formulas, such as the discriminant method, to analyze or solve the equation.
For example, the equation \( x^2 = 6x - 9 \) must be rearranged by moving all terms to one side of the equation, providing a clear signature of the quadratic equation with \( a \), \( b \), and \( c \) defined:

• First, we move all terms to one side, usually the left: \( x^2 - 6x + 9 = 0 \).
• Here, \( a = 1 \), \( b = -6 \), and \( c = 9 \).

Writing equations in this form streamlines the process of solving and analyzing them, making it easier to apply mathematical rules and operational methods.