Solving for Two Numbers That Multiply to 45 and Add to -14: A Comprehensive Guide
Algebra presents various challenges, but one of the most common and valuable exercises involves finding two numbers that satisfy both multiplicative and additive conditions. In this guide, we'll explore how to solve the equation where two numbers multiply to 45 and add to -14. We'll present the solution using both the factor method and the quadratic equation, ensuring a clear and comprehensive understanding.
Setting Up the Equations
Let's denote the two numbers by x and y. The given conditions are:
x y 45 x y -14Solving Using the Factor Method
We can express y in terms of x from the second equation:
y -14 - x
Substitute this expression for y into the first equation:
x (-14 - x) 45
Expanding this gives:
-x2 - 14x 45
Rearranging the equation leads to a quadratic equation:
x2 - 14x - 45 0
Factoring the Quadratic Equation
To factor the quadratic equation x2 - 14x - 45 0, we need to find two numbers that multiply to -45 and add to -14. The numbers that fit this requirement are -9 and -5:
x2 - 9x - 5x - 45 0 (x - 9) (x - 5) 0
Setting each factor to zero gives us:
x - 9 0 x 9 x - 5 0 x 5Therefore, the two numbers are x -5 and x -9 (since we are dealing with the negative values).
Verification
To verify our solution, we can substitute these values back into the original conditions:
Multiplication: -5 -9 45 Addition: -5 -9 -14The solutions satisfy both conditions, confirming that -5 and -9 are the correct numbers.
Solving Using the Quadratic Formula
We can also solve the quadratic equation x2 - 14x - 45 0 using the quadratic formula:
x frac{-b pm sqrt{b^2 - 4ac}}{2a}
Here, a 1, b -14, and c -45. Plugging these values in, we get:
x frac{-(-14) pm sqrt{(-14)^2 - 4 cdot 1 cdot (-45)}}{2 cdot 1}
x frac{14 pm sqrt{196 180}}{2}
x frac{14 pm sqrt{376}}{2}
x 7 pm sqrt{94}
However, since the solutions need to be integers, we can verify that the simpler factor method yielded the correct integer solutions of -5 and -9.
Conclusion
The two numbers that multiply to 45 and add to -14 are -5 and -9. This guide demonstrates both the factor method and the quadratic equation approach, ensuring a thorough understanding of the solution process.
Additional Resources
For further practice, you may want to explore similar algebraic problems and develop a stronger grasp of quadratic equations. Online resources and practice tests are readily available to help refine your algebraic skills.