Solving Quadratic Equation: Find A, B, C With Roots -4/7 & 5
Have you ever wondered how to construct a quadratic equation if you know its roots? Let's dive into a fascinating problem where we'll do just that! We'll explore how to determine the coefficients of a quadratic equation given its solutions. This involves understanding the relationship between the roots and coefficients, a key concept in algebra. Let's break down the problem step by step, making it easy to follow and understand.
Understanding the Problem
In this mathematical puzzle, William is thinking about a quadratic equation, which can be generally represented as ax^2 + bx + c = 0. Here's what we know:
- a is a positive integer less than 10. This means a can be any whole number from 1 to 9.
- b and c are integers. Integers include all whole numbers (positive, negative, and zero).
- The two solutions (also called roots) to William's equation are x = -4/7 and x = 5.
Our mission is to find the specific values of a, b, and c that satisfy these conditions. This isn't just about plugging in numbers; it's about understanding the underlying structure of quadratic equations and how their roots relate to their coefficients.
Why This Matters
Understanding quadratic equations is crucial in various fields, from physics and engineering to economics and computer science. They model many real-world phenomena, such as the trajectory of a projectile, the shape of a suspension bridge, and the growth of populations. Being able to solve and manipulate quadratic equations is a fundamental skill in mathematics and beyond. The process we'll use to solve this problem highlights several important mathematical concepts, including the relationship between roots and factors, the manipulation of algebraic expressions, and the importance of integer constraints. By working through this problem, we're not just finding the solution; we're building a deeper understanding of these concepts.
Building the Equation from the Roots
The cornerstone of solving this problem lies in the connection between the roots of a quadratic equation and its factored form. If we know the roots, we can construct the factors, and from the factors, we can build the equation. This is a powerful technique that simplifies the process of finding the equation.
From Roots to Factors
Since x = -4/7 and x = 5 are the solutions, we can work backward to find the factors. Remember that if x = r is a solution to an equation, then (x - r) is a factor of the equation. Applying this:
- For x = -4/7, the factor is (x - (-4/7)), which simplifies to (x + 4/7).
- For x = 5, the factor is (x - 5).
So, our quadratic equation can be written in factored form as:
(x + 4/7)(x - 5) = 0
This equation has the correct roots, but it's not quite in the form we need yet. We need to get rid of the fraction and ensure that the coefficients are integers.
Eliminating the Fraction
To get rid of the fraction, we can multiply the entire equation by 7. This gives us:
7(x + 4/7)(x - 5) = 0
Distributing the 7 into the first factor, we get:
(7x + 4)(x - 5) = 0
Now we have an equation with integer coefficients, which is closer to the form we're looking for.
Expanding and Simplifying
Now that we have the equation in factored form with integer coefficients, the next step is to expand it and simplify it into the standard quadratic form ax^2 + bx + c = 0. This involves multiplying the two factors and combining like terms.
Expanding the Product
Let's expand the product (7x + 4)(x - 5) using the distributive property (often remembered by the acronym FOIL - First, Outer, Inner, Last):
- First: 7x * x = 7x^2
- Outer: 7x * -5 = -35x
- Inner: 4 * x = 4x
- Last: 4 * -5 = -20
Combining these terms, we get:
7x^2 - 35x + 4x - 20 = 0
Simplifying the Equation
Now, let's combine the like terms (the x terms) to simplify the equation:
7x^2 + (-35x + 4x) - 20 = 0
7x^2 - 31x - 20 = 0
Now our equation is in the standard quadratic form ax^2 + bx + c = 0.
Identifying a, b, and c
With the equation in the standard form, it's straightforward to identify the values of a, b, and c. We simply compare our equation to the general form and read off the coefficients.
Matching Coefficients
Our equation is:
7x^2 - 31x - 20 = 0
The general form is:
ax^2 + bx + c = 0
By comparing the coefficients, we can see:
- a = 7
- b = -31
- c = -20
Verifying the Solution
Let's double-check that our solution makes sense. We know that a must be a positive integer less than 10, which 7 satisfies. We also know that b and c must be integers, which -31 and -20 satisfy. Finally, the most critical verification step is to see if these values, when plugged back into the quadratic formula or by factoring, yield the original roots of x = -4/7 and x = 5. This confirms that our solution is correct.
The Quadratic Formula
As an additional check, we can use the quadratic formula to find the roots of the equation 7x^2 - 31x - 20 = 0. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in our values for a, b, and c:
x = (31 ± √((-31)^2 - 4 * 7 * -20)) / (2 * 7)
x = (31 ± √(961 + 560)) / 14
x = (31 ± √1521) / 14
x = (31 ± 39) / 14
This gives us two solutions:
- x = (31 + 39) / 14 = 70 / 14 = 5
- x = (31 - 39) / 14 = -8 / 14 = -4/7
These match our original roots, confirming our solution.
Conclusion
We've successfully navigated through William's equation and found the values of a, b, and c. By understanding the relationship between the roots and the factored form of a quadratic equation, we were able to construct the equation and identify the coefficients. This exercise highlights the power of algebraic manipulation and the importance of understanding fundamental mathematical concepts. In summary:
- a = 7
- b = -31
- c = -20
This problem is a great illustration of how mathematical principles can be applied to solve intriguing puzzles. From understanding the nature of quadratic equations to applying the quadratic formula as a verification step, the process showcases the elegance and interconnectedness of mathematical concepts. For further exploration of quadratic equations and their properties, you might find the resources at Khan Academy helpful.
