Problem 81 - Elementary Algebra

Question

Find a quadratic equation with integer coefficients, given the following solutions. $$ -3,1 $$

Step-by-Step Solution

Answer

The quadratic equation is \( x^2 + 2x - 3 = 0 \).

Step 1: Understand the Roots

We have two roots of the quadratic equation: \( x = -3 \) and \( x = 1 \). These roots correspond to the values of \( x \) where the quadratic equation evaluates to zero.

Step 2: Use the Roots to Form Factors

The quadratic equation can be expressed in terms of its roots. For a root \( r \), the corresponding factor is \( (x - r) \). So, the factors for our roots are \( (x + 3) \) and \( (x - 1) \).

Step 3: Construct the Quadratic Equation

To form the quadratic equation, multiply the factors together: \[ (x + 3)(x - 1) = 0 \] Expand the expression to achieve the quadratic form.

Step 4: Expand the Expression

Apply the distributive property to expand the product: \[ (x + 3)(x - 1) = x^2 - x + 3x - 3 \] Combine like terms to simplify: \[ x^2 + 2x - 3 = 0 \]

Step 5: Verify Integer Coefficients

Confirm that the quadratic equation \( x^2 + 2x - 3 = 0 \) has integer coefficients (1, 2, and -3), ensuring it meets the problem's requirements.

Key Concepts

Integer Coefficients

When working with quadratic equations, integer coefficients play a vital role. These are the numbers placed in front of variables, and they're always whole numbers like -3, 0, 2, or 15. Having integer coefficients in a quadratic equation simplifies computation and makes the equation easier to interpret. For instance, in the equation

• \(x^2 + 2x - 3 = 0\)

all coefficients—1 (for \(x^2\)), 2 (for \(x\)), and -3—are integers. Working with integers is straightforward, and often these numbers help when factoring or finding roots.

Roots of a Quadratic

The roots of a quadratic equation are the values of x that make the equation true. Essentially, they're the 'solutions' to the equation. For the given exercise, the roots were provided as -3 and 1. This means that substituting these values for x in the quadratic equation results in zero:

• \(x = -3\) gives zero when plugged into the equation
• \(x = 1\)

Knowing the roots lets you construct the equation. Each root gives rise to a factor of the form \((x - r)\), where r is a root. Hence, the equation formed has factors \((x + 3)\) and \((x - 1)\). Understanding this relationship is key to moving from solutions to a full quadratic equation.

Expanding Expressions

Expanding expressions from their factored form involves breaking down and simplifying the algebraic terms. In our exercise, you're given

• \((x + 3)(x - 1)\)

To expand this, apply algebraic principles to reach the standard form of a quadratic equation:

• \(x \times x\) gives \(x^2\)
• x multiplied by -1 gives -x
• 3 multiplied by x gives +3x
• 3 multiplied by -1 gives -3

Adding these terms together results in the expanded expression \(x^2 + 2x - 3\). This expanded form illustrates the complete picture of a quadratic equation and is crucial for graphing or further solving.

Distributive Property

The distributive property is a fundamental concept in algebra used to expand expressions. It states that a term multiplied by a sum of terms inside a bracket is distributed to each term within the bracket. Formally, it is expressed as:

• \(a(b + c) = ab + ac\)

In our quadratic problem, when expanding

• \((x + 3)(x - 1)\)

we distribute x and 3 over \((x - 1)\). This gives:

• \(x(x - 1) + 3(x - 1)\)

Breaking it down further shows the step-by-step application of the property, resulting in the equation:

• \(x^2 - x + 3x - 3\)

Finally, combining like terms uses the distributive property to form a complete and simplified quadratic expression. This approach is essential for anyone seeking to understand or solve more complex algebraic equations.