Problem 93 - Elementary Algebra

Question

Find a quadratic equation with integer coefficients, given the following solutions. $$ 3, \text { double root } $$

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 - 6x + 9 = 0\).

Step 1: Understand the Problem

The problem asks for a quadratic equation with integer coefficients that has solutions of 3 as a double root, meaning the equation has one root that is equal to 3, and it is repeated.

Step 2: Use the Concept of Double Root

If a quadratic equation has a double root, this means the root appears twice. For a root \( r \), the factor of the quadratic would be \((x - r)^2\). In this case, since the root is 3, the factor is \((x - 3)^2\).

Step 3: Expand the Factor

Expand the expression \((x - 3)^2\) to find the quadratic equation. This gives:\[(x - 3)(x - 3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9\].

Step 4: Write the Quadratic Equation

The expanded expression is your quadratic equation with integer coefficients: \(x^2 - 6x + 9 = 0\). This has the required double root of 3.

Key Concepts

Double Roots

In algebra, the term "double root" refers to a situation where a quadratic equation has a repeated solution. This means the same number satisfies the equation twice. A common notation for a root is 'r'. When a quadratic equation has a double root, it can be expressed in the format

• \( (x - r)^2 = 0 \)

This form signifies that the root \( r \) is repeated.
In the given problem, the double root is the number 3. So, the root appears as a factor twice, represented as

• \( (x - 3)^2 \)

A quadratic equation with a double root is unique because it touches the x-axis at one point, representing the root. This touch point indicates the parabola formed by the equation only "kisses" the x-axis at that spot, rather than crossing it.

Integer Coefficients

When creating a quadratic equation, integer coefficients are often required. Coefficients are the numbers that multiply the variables in the equation. In the quadratic equation format,

• \( ax^2 + bx + c = 0 \)

the letters \( a \), \( b \), and \( c \) are the coefficients. For them to be integers, they must be whole numbers, whether positive or negative, such as -2, 0, or 5.
Integer coefficients ensure the equation remains simple and devoid of fractions or decimals, making calculations easier. In our problem, after expanding the expression \((x - 3)^2\), the coefficients are all integers:

• The x-squared term has a coefficient of 1
• The x term has a coefficient of -6
• The constant term is 9

Thus, the equation \( x^2 - 6x + 9 = 0 \) is presented with integer coefficients.

Expanding Expressions

Expanding expressions is a fundamental algebraic process that involves multiplying terms to eliminate any brackets. This helps to transform an equation into a standard form.
For a quadratic equation with a double root, like \((x - 3)^2\), expanding is crucial to express the equation in terms of \( ax^2 + bx + c \). When expanding

• \( (x - 3)(x - 3) \)

we apply the distributive property, multiplying each term of the first binomial by each term of the second:

• Multiply \( x \) by \( x \) to get \( x^2 \)
• Multiply \( x \) by -3 to get \(-3x \)
• Multiply -3 by \( x \) to get another \(-3x \)
• Multiply -3 by -3 to get 9

Adding these products results in \[ x^2 - 3x - 3x + 9 \],
which simplifies to \( x^2 - 6x + 9 \). This expansion process is a mechanical yet essential step to formulating a clear quadratic equation.