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Chapter 10: Problem 53

Write a quadratic equation with the given solutions. \(-5\) and \(-6\)

Short Answer

Expert verified
The quadratic equation with solutions -5 and -6 is \(x^2 + 11x + 30 = 0\).

Step by step solution

01

Calculate the sum of the solutions

First, calculate the sum of the solutions. This is \(a+b\), where \(a\) and \(b\) are the solutions of the equation. In this case, \(a=-5\) and \(b=-6\), so \(a+b=-5+(-6)=-11\).
02

Calculate the product of the solutions

Next, calculate the product of the solutions. This is \(ab\), where \(a\) and \(b\) are the solutions of the equation. So \(ab=-5*(-6)=30\).
03

Substitute into the equation

Now substitute both the sum and the product of the solutions into the quadratic equation, which is \(x^2 - (a+b)x + ab = 0\). From the previous steps, we found that \(a+b=-11\) and \(ab=30\). Substituting in these values, we get \(x^2 - (-11)x + 30 = 0\), which simplifies to \(x^2 + 11x + 30 = 0\). This is the quadratic equation with solutions -5 and -6.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Sum of Solutions
In quadratic equations, understanding the sum of solutions can be a helpful tool. When you have the solutions (or roots) of a quadratic equation, such as
  • \(a\) and \(b\), the solutions of the equation
  • Simply add them together to find their sum
  • The formula used is \(a + b\)
In this specific exercise, the solutions are \(-5\) and \(-6\). Therefore, the sum is:\[-5 + (-6) = -11\]This sum is a key parameter for constructing the quadratic equation. It determines the coefficient of the \(x\) term in the standard form such as \(x^2 - (a+b)x + ab = 0\).
Understanding how the sum influences this position helps simplify expressing your equation correctly. Always pay attention to the sign! This detail ensures that the correct form of the equation is achieved when substituting.
Exploring the Product of Solutions
Whenever dealing with quadratic equations, calculating the product of solutions is central to forming the equation. The product of the solutions, denoted by \(ab\), directly determines another part of the equation:

  • Multiply the solutions together: \(-5 imes (-6)\)
  • The result here is 30
The product is placed as the constant term in the quadratic equation. This relationship is vital, especially since the correctness of the equation relies heavily on accurate computation.
Remember: Each multiplication of solutions affects the structure and appearance of the equation.
Forming an equation like \(x^2 - (a+b)x + ab = 0\) with correct products simplifies solving and understanding further mathematical or real-world scenarios.
Solution Substitution in Quadratic Equations
By substituting calculated sums and products of the solutions into a quadratic equation, you craft an equation that reflects these solutions perfectly. Here is how substitution works with our calculations:
  • Start with the basic form \(x^2 - (a+b)x + ab = 0\)
  • Replace \(a+b\) with \(-11\)
  • Replace \(ab\) with 30
Substituting these values into the equation, you yield:\[x^2 - (-11)x + 30 = 0\]Simplify this to:\[x^2 + 11x + 30 = 0\]This substitutive approach makes the creation of the equation straightforward.
It combines all the key parts and parameters found through earlier steps, transforming theoretical calculations into a concrete quadratic equation.
This process is crucial for accurately identifying the equation with desired roots.

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Most popular questions from this chapter

Find the product. $$(3 x+5)(3 x+5)$$

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