Problem 44 - Algebra A Combined Function

Question

Use the discriminant to determine the number and types of solutions of each equation. $$ 9 x^{2}+1=6 x $$

Step-by-Step Solution

Answer

The quadratic equation has one real repeated solution.

Step 1: Rewrite the equation in standard form

Start by bringing all terms to one side of the equation to have the quadratic equation in standard form, which is \( ax^2 + bx + c = 0 \). \( 9x^2 + 1 = 6x \) becomes \( 9x^2 - 6x + 1 = 0 \).

Step 2: Identify the coefficients

In the quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients: \( a = 9 \), \( b = -6 \), and \( c = 1 \).

Step 3: Calculate the discriminant

The discriminant \( \Delta \) of a quadratic equation is found using the formula \( \Delta = b^2 - 4ac \). Substitute the identified coefficients into the formula: \( \Delta = (-6)^2 - 4 \times 9 \times 1 \).

Step 4: Simplify the discriminant

Calculate the value inside the discriminant: \( (-6)^2 = 36 \) and \( 4 \times 9 \times 1 = 36 \). Thus, \( \Delta = 36 - 36 = 0 \).

Step 5: Determine the number and type of solutions

When the discriminant \( \Delta = 0 \), there is exactly one real and repeated (or double) solution. This means the quadratic equation has a perfect square trinomial and the graph will touch the x-axis at just one point without crossing it.

Key Concepts

Discriminant

The discriminant is an essential part of solving quadratic equations when using the quadratic formula. It is a key component in understanding the characteristics of the solutions without actually solving the equation. To find the discriminant, you use the formula \( \Delta = b^2 - 4ac \). You obtain this by identifying the coefficients \( a \), \( b \), and \( c \) from a quadratic equation in standard form: \( ax^2 + bx + c = 0 \). Once you substitute the coefficients into the formula, it reveals how many and what type of solutions the quadratic equation has.
For instance, in our exercise example, the equation becomes \( 9x^2 - 6x + 1 = 0 \) leading to coefficients \( a = 9 \), \( b = -6 \), and \( c = 1 \). Calculating the discriminant gives \( \Delta = (-6)^2 - 4 \times 9 \times 1 = 0 \). Thus, this calculation is crucial as it indicates the nature of the roots without solving for them directly.

Standard Form

Quadratic equations are most straightforward to analyze when written in standard form, which is \( ax^2 + bx + c = 0 \). In this format, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). Converting an equation to this form makes it easier to apply methods like the quadratic formula and to calculate the discriminant.
To transform the equation into standard form, first move all terms to one side of the equation, so the other side equals zero. For example, the original equation \( 9x^2 + 1 = 6x \) was rewritten as \( 9x^2 - 6x + 1 = 0 \). This adjustment clarifies the role of each term in the equation and sets the stage for further calculations like identifying the discriminant.

Types of Solutions

Understanding the types of solutions for a quadratic equation comes from analyzing the discriminant.

• If \( \Delta > 0 \), the equation has two distinct real solutions, meaning the parabola intersects the x-axis at two points.
• If \( \Delta = 0 \), there is one real double root, indicating the parabola touches the x-axis at exactly one point, without crossing.
• If \( \Delta < 0 \), the equation has no real solutions but two complex conjugate solutions, so the parabola does not intersect the x-axis at all.

In the given exercise, with a discriminant \( \Delta = 0 \), the equation has one real, repeated solution. This situation signifies the x-axis is simply touched by the parabola at a single point. Such an insight helps visualize the behavior of the quadratic expression graphically as well.