Algebra Know-It-ALL

390 Quadratic Equations with Complex Roots

which simplifies to

2 x^2 + 79 = 0

Here’s a challenge!

Investigate what happens in the general case if a is positive and c is negative in the quadratic equation

ax^2 +c= 0

Solution

We have a> 0 and c< 0. That means −c> 0. Let’s rewrite the above equation as

ax^2 − (−c)= 0

We can add −c to each side, getting

ax^2 =−c

Dividing through by a, we obtain

x^2 =−c/a

Because −c> 0 and a> 0, we know that −c/a> 0. We can take the positive-negative square root of both

sides to get

x=±(−c/a)1/2

Stated separately, the roots are

x= (−c/a)1/2 or x=−[(−c/a)1/2]

These are both real numbers, and are additive inverses.

Here’s another challenge!

Investigate what happens in the general case if a is negative and c is positive in the quadratic equation

ax^2 +c= 0

Solution

We have a< 0 and c> 0. That means −a> 0. We can subtract c from each side, getting

ax^2 =−c

Multiplying through by −1 gives us

(−a)x^2 =c