How to Solve Quadratic Equations with One Known Root

When solving quadratic equations in the form of (y ax^2 bx c), you might be given a root, making the process more manageable. This article will guide you through the steps to find the unknown root of a quadratic equation when one of the roots is known.

Understanding the Basics

A quadratic equation in standard form is given by (y ax^2 bx c), where (a), (b), and (c) are constants. If you are told that (x 3) is a root of the quadratic, it means that substituting (x 3) will make the value of the quadratic equal to zero:

(y a(3)^2 b(3) c 9a 3b c 0)

However, a single equation with three unknowns (variables) is not sufficient to solve for all three constants. You need two of the constants to move forward. This is where the known root becomes valuable.

Using a Known Root to Find the Unknowns

Suppose the quadratic equation is given as (y kx^2 - x) or (y x^2 - kx) or (y x^2 - x - k), where (k) is a known constant. Substituting the known root, say (x 3), into the equation will give you a single equation with one unknown, allowing you to solve for the other constants.

Example Walkthrough

Lets consider an example where the equation is (y x^2 - kx - 6) and (x -2) is a root. To find (k), we substitute (x -2) into the equation:

((-2)^2 - k(-2) - 6 0)

Simplifying this, we get:

(4 2k - 6 0)

(2k - 2 0)

(2k 2)

(k 1)

Now that we know (k 1), the quadratic equation becomes (y x^2 - x - 6). We can factorize this equation further:

(x^2 - x - 6 (x - 2)(x 3))

This gives us the roots (x 2) and (x -3). However, we already know one root is (x -2), so the complete set of roots is (x -2) and (x -3).

Conclusion

When solving quadratic equations with one known root, the process is streamlined. By substituting the known root into the equation, you can derive a single linear equation involving the unknown constants. Solving these linear equations enables you to find the unknown constants and hence the other root(s) of the quadratic equation.

This method is particularly useful in many practical applications, such as physics, engineering, and finance, where quadratic equations often appear. By mastering this technique, you will be better equipped to tackle more complex problems.

Keywords: quadratic equations, known root, unknown root, solving quadratic equations