Solving Quadratic Equations Without a B Term: Simplified Steps and Use Cases

When faced with a quadratic equation that lacks the b term, such as ax2 c 0, many students and professionals often question whether the standard quadratic formula can still be applied. The good news is that the absence of the b term simplifies the process and does not preclude the use of the quadratic formula.

Understanding the Quadratic Equation Without a B Term

Consider the quadratic equation:

ax2 c 0

This equation can be simplified as follows:

Subtract c from both sides:

ax2 -c

Divide both sides by a:

x2 -c/a

Take the square root of both sides:

x ±√(-c/a)

This results in the exact root solutions using the fundamental principles of algebra.

Application of the Quadratic Formula

The quadratic formula, which is a powerful tool for solving quadratic equations, is:

x [-b ± √(b2 - 4ac)] / (2a)

When the b term is zero, the formula simplifies significantly:

x ±√(-c/a)

This simplification is valid as long as a is non-zero and c is a real number. Here’s an example to illustrate:

x2 - 1 0

Using the quadratic formula (with b 0):

x ±√[-4 * a * -1] / (2a) ±√(4) / 2 ±2 / 2 ±1

Thus, the roots are:

x 1 x -1

Further Examples and Generalizations

Consider the general form of the quadratic equation:

ax2 bx c 0

If b 0, the equation becomes:

ax2 c 0

Applying the quadratic formula with b 0:

x [-0 ± √(02 - 4ac)] / (2a) ±√(-4ac) / (2a) ±√(-c/a)

This confirms that the roots are:

x ±√(-c/a)

For the roots to be real, either a c

Conclusion

While the presence of the b term in a quadratic equation is often seen as a necessary condition for applying the quadratic formula, an equation lacking b is still completely solvable using algebraic techniques. The absence of b simplifies the process, making it easier to find the roots.