Problem 131 - College Algebra

Question

Write a quadratic equation in general form whose solution set is {-3,5}.

Step-by-Step Solution

Answer

The quadratic equation in general form with solutions -3 and 5 is \(x^2 - 2x - 15 = 0\).

Step 1: Define the Roots

The roots of the equation have been provided as -3 and 5.

Step 2: Insert the Roots into the Factored Form

We can substitute these roots into the factored form of a quadratic equation. Let the coefficient 'a' equal to 1, we get \( (x + 3)(x - 5) = 0\).

Step 3: Expand to Obtain General Form

We expand the equation to obtain the equation in standard form: \(x^2 - 5x + 3x - 15 = 0 \), which simplifies to \(x^2 - 2x - 15 = 0\).

Key Concepts

Factored Form

The factored form of a quadratic equation is a way of expressing the equation as a product of its linear factors. This is written as \[ (x - p)(x - q) = 0 \] where \( p \) and \( q \) are the roots of the equation.
To find the factored form, you need the roots of the equation. These roots are simply the values of \( x \) that make the equation equal to zero.
In our example, the roots are given as -3 and 5, so the factored form becomes \[(x + 3)(x - 5) = 0\].
The factored form gives you a direct insight into the roots and is very useful for solving quadratic equations as it breaks down the equation into simpler parts.

Roots of a Quadratic Equation

The roots of a quadratic equation, sometimes called solutions or zeros, are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \).
These roots can be real or complex numbers.
For quadratic equations, the roots can be found using a variety of methods such as factoring, completing the square, or utilizing the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
In our particular problem, the solutions were stated directly as -3 and 5.
Knowing the roots allows us to easily write the factored form of the equation, which can then be expanded or manipulated to suit different purposes.

General Form of a Quadratic Equation

The general form of a quadratic equation is represented as \[ ax^2 + bx + c = 0 \].
It is the standard way to express quadratics and is useful for a range of different mathematical applications including graphing and solving.
Starting from the factored form, such as \[ (x + 3)(x - 5) = 0\], we can expand it to move to the general form.
This involves multiplying the factors together, which gives us \[ x^2 - 5x + 3x - 15 = 0 \], then combining like terms leads to \[ x^2 - 2x - 15 = 0 \].
This is now the general form of the quadratic equation with coefficients \( a = 1 \), \( b = -2 \), and \( c = -15 \). Understanding how to transition between different forms of a quadratic equation can greatly enhance problem-solving capabilities.