A pure quadratic equation is an quadratic equation that has no linear term. So we can say that a pure quadratic equation is any equation in the form

^{2}+ c = 0, where a, c ∈ ℝ and a≠0.

Why is it so special? Because it’s really easy to solve this equation. Suppose this equation:

^{2}− 18 = 0

How should we solve it? First, we add * + 18*:

2x^{2} − 18 | = | 0 |

2x^{2} | = | 18 |

The quadratic term is alone now. We divide by *2*:

2x^{2} | = | 18 |

x^{2} | = | 9 |

Now we ask ourself a question: *what number should we squared to get nine?* There are two numbers that satisfy this condition: *3* because *3 ^{2} = 9* and the other number is

*− 3*since

*( − 3)*also equals

^{2}*9*. We can write down by extracting the equation:

Take a look in the graph of the function *f(x) = 2x ^{2} − 18*. It intersects the horizontal axis exactly in points

*( − 3, 0)*and

*(0, 3)*.

## The General Solution

We can infer the general approach how to solve any pure quadratic equation. Let’s have an equation

^{2}+ c = 0

We just subtract the constant *c*:

ax^{2} + c | = | 0 |

ax^{2} | = | − c |

Now we divide the equation by *a*:

The final step is extracting the root:

Now we have to distinguish three possibilities:

**The result**The equation has solution and it has exactly two roots. First one is*− (c/a)*is positive.*− (c/a)*and the second one is*(c/a)*.**The result**The equation has a solution, there is just a single root since*− (c/a)*equals zero.*− (c/a) = c/a = 0***The result**In this case the equation doesn’t have any solution since we cannot extract the root from negative number..*− (c/a)*is negative.

## Examples

**Example 1:** Let’s go back to our first equation: *2x ^{2} − 18 = 0*. What are the coefficients? It’s

*a = 2*and

*c = − 18*. We can use the formula

to compute roots of the equation. We substitute coefficients *a* and *c*:

The roots of this equation are *x _{1} = − 3* and

*x*.

_{2}= 3**Example 2:** Solve the equation

Ok, so first we need to identify the coefficients. We can actually rewrite the equation this way:

Now it’s clear that the quadratic coefficients equals *a = 1/3*. The constant is not there thus it equals zero, *c = 0*. We can now use the magic formula:

We hit the special case when we have just a single root, *x = 0*. We can draw a graph of the function *f(x) = x ^{2}/3*:

From the graph it’s easy to see why is there just a single root: the graph intersects with the horizontal axis in just a single point *(0, 0)*.

**Example 3:** The last example:

^{2}+ 2 = 0.

Now we have *a = 1* and *c = 2*. We use the magic formula again:

The expression * − (c/a)* is negative and we cannow compute square root of a negative number. Thus this equation doesn’t have any solution. We can see it from the graph since it doesn’t intersects the horizontal axis at all: