Question

Write a quadratic equation with the given solutions. \(-41\) and 5

Short answer

The quadratic equation with the roots -41 and 5 is \(x^2 + 36x - 205 = 0\).

Step-by-step solution

- Recognize the Mathematical Property

Understand that the property of a quadratic equation states that if the roots of the equation \(ax^2 + bx + c = 0\) are \(p\) and \(q\), then \(p+q=-\frac{b}{a}\), and \(pq=\frac{c}{a}\). This is because in a factorized form, a quadratic equation looks like this: \(a(x - p)(x - q) = 0\).

- Identify Given Roots

According to the problem, the roots have been identified as -41 and 5. Hence, the equation can be expressed as \(a(x + 41)(x - 5) = 0\), which is the formula built using the mathematical property just described in step 1.

- Formulate the Quadratic Equation

Upon distributing through, our quadratic equation looks like this: \(a(x^2 + 36x - 205) = 0\), or \(x^2 + 36x - 205 = 0\), assuming \(a = 1\). This is the quadratic equation to represent the roots -41 and 5.