Problem 89 - Elementary Algebra

Question

Find a quadratic equation with integer coefficients, given the following solutions. $$ -1 / 2,1 / 2 $$

Step-by-Step Solution

Answer

The quadratic equation is \(4x^2 - 1 = 0\).

Step 1: Write the roots in fractional form

The given solutions to the quadratic equation are \(-\frac{1}{2}\) and \(\frac{1}{2}\). These fractions represent the roots of the quadratic equation.

Step 2: Convert the fractions into linear factors

Express each root as a linear factor. For the root \(-\frac{1}{2}\), the factor is \((2x + 1)\). Similarly, for the root \(\frac{1}{2}\), the factor is \((2x - 1)\).

Step 3: Form the quadratic equation by multiplying the factors

Multiply the two linear factors to obtain a quadratic equation:\[(2x + 1)(2x - 1) = 0\]This expands to:\[4x^2 - 1 = 0\]

Step 4: Verify the equation has integer coefficients

On expanding, we obtain \(4x^2 - 1 = 0\). This equation has integer coefficients (namely, 4, 0, and -1), which is valid as required.

Key Concepts

Integer Coefficients

When we talk about quadratic equations with integer coefficients, it simply means that the numbers multiplying each term of the equation are whole numbers without fractions or decimals. Consider a general quadratic equation, usually written as:\[ ax^2 + bx + c = 0 \]Here, \(a\), \(b\), and \(c\) are the coefficients. For an equation to have integer coefficients, \(a\), \(b\), and \(c\) must be integers. In our worked example, we obtained the equation:\[ 4x^2 - 1 = 0 \]The coefficients here are 4 for \(x^2\), 0 for \(x\), and -1 for the constant term. Because these are all integers, the equation meets the requirement for integer coefficients.

• Ensure all terms of your quadratic equation are devoid of fractions.
• Multiply through by common denominators if you start with fractional roots.

Integer coefficients often make quadratic equations simpler to solve using methods like factoring.

Factorization

Factorization is a process of breaking down an expression into a product of simpler expressions, or factors, that when multiplied together give the original expression. In quadratic equations, this technique is essential.To factorize a quadratic equation, you often look for two numbers that add up to \(b\) (the coefficient of \(x\)) and multiply to give \(ac\) (the product of the coefficient of \(x^2\) and the constant term) if the equation is in standard form \(ax^2 + bx + c\). In our example, when roots are given, factorization involves creating binomial expressions from these roots. Specifically:1. Using root \(-\frac{1}{2}\) gives the factor \((2x + 1)\).2. Using root \(\frac{1}{2}\) gives the factor \((2x - 1)\).By multiplying these factors:\[(2x + 1)(2x - 1)\] we reconstruct the quadratic equation:\[4x^2 - 1 = 0\]

• Factorization simplifies solving equations by reducing them to linear expressions.
• Helps in identifying roots directly for integer coefficients.

Roots of Equations

The roots of an equation are the values of \(x\) that satisfy the equation, making it true. For quadratic equations, these can be real numbers or complex numbers, and are often found using methods such as factorization, completing the square, or using the quadratic formula.When given roots in an exercise, like \(-\frac{1}{2}\) and \(\frac{1}{2}\) for our example, these indicate positions on the number line where the equation crosses the \(x\)-axis, meaning the output of the equation is zero at these points. To construct a quadratic equation from these roots:

• Write each root as a linear expression set equal to zero; e.g., for root \(-\frac{1}{2}\), write \(2x + 1 = 0\).
• Do the same for root \(\frac{1}{2}\) as \(2x - 1 = 0\).
• Multiply these linear factors to revert back to quadratic form, as shown in the example.

Understanding and working with roots allows you to reverse-engineer an equation, providing insight into the range and behavior of quadratic functions.