Math FoundationsEquationsQuadratic Equations

Incomplete Quadratic Equations

Quadratic equations with missing terms (incomplete quadratics) are the simplest forms of quadratic equations where the coefficient B, C, or both are missing. Let's learn how to solve them correctly in each specific case.
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What is it?

You have already met and learned to recognize quadratic equations. Now it is time to learn how to solve them. Before tackling full quadratic equations of general form, let's start with something simpler. Sometimes, a quadratic equation is missing some parts of the general form. Such quadratic equations are called incomplete.

Incomplete quadratic equation

Quadratic equation, in which coefficient B or C or both are zero:
Examples:
10x2=0⏟B=0 and C=0\underbrace{10x^2 = 0}_{B = 0 \ \text{and} \ C = 0}
x2+x=0⏟C=0 \underbrace{x^2 + x = 0}_{C = 0}
3x2−8=0⏟B=0 \underbrace{3x^2 - 8 = 0}_{B = 0}
What if coefficient A is zero? That can happen too, but then, by definition, it won't be a quadratic equation anymore, because the thing that makes it “quadratic” — x2x^2 — will disappear!

Solving incomplete quadratic equations

The good news is that any of the three types of incomplete quadratic equations is solved very simply, without using any tricks or complicated calculations. Let's analyze each case separately.

If both “B” and “C” are zero

The simplest of all three types of incomplete quadratic equations — when both coefficients B and C are zero:
8x2=08\brand{x^2} = 0
−72x2=0 -\frac{7}{2}\brand{x^2} = 0
2+log⁡29999−563x2=0 \frac{\sqrt{2} + \log_2{9}}{999 - 56^3}\brand{x^2} = 0
It absolutely doesn't matter what stands next to x2x^2. The point is that this “something” is multiplied by our unknown, and the result must be zero. How do we get zero with 100% guarantee during multiplication? Very simple: zero will result if the unknown itself is zero! The equation always has a unique solution:
8⋅0=0x=08 \cdot \brand{0} = 0 \\ \boxed{x = 0}
−72⋅0=0x=0 -\frac{7}{2} \cdot \brand{0} = 0 \\ \boxed{x = 0}
2+log⁡29999−563⋅0=0x=0 \frac{\sqrt{2} + \log_2{9}}{999 - 56^3} \cdot \brand{0} = 0 \\ \boxed{x = 0}

Quadratic equation root when “B” = 0 and “C” = 0

Any incomplete quadratic equation with zero coefficients B and C always has a unique solution x = 0 and it doesn't matter what coefficient A is!
Proof

If “C” is zero

The next type of incomplete quadratic equations — when coefficient C is zero. To solve them, it's enough to master the same action rule and the ability to solve equations using the set of factors equal to zero.

Quadratic equations when “C” = 0

😀
Elementary
Solve the equation:
x2+5x=0x^2 + 5x = 0

Quadratic equation roots when “C” = 0

Any incomplete quadratic equation with zero coefficient C always has two roots, which can be found by formulas:
Proof
The benefit of the general formula is that you don't need to factor out and do other transformations every time. It suffices to just look at the equation, determine coefficients A and B and immediately substitute them into the formula.
0=x+x2x1=0x2=−11=−10 = x + x^2 \\ \boxed{x_1 = 0} \\ \boxed{x_2 = -\frac{1}{1} = -1}
−16x=4x2x1=0x2=−−164=4 -16x = 4x^2 \\ \boxed{x_1 = 0} \\ \boxed{x_2 = -\frac{-16}{4} = 4}
12x2+20x=0x1=0x2=−2012=−201⋅21=−40 \frac{1}{2}x^2 + 20x = 0 \\ \boxed{x_1 = 0} \\ \boxed{x_2 = -\frac{20}{\frac{1}{2}} = -\frac{20}{1}\cdot\frac{2}{1} = -40}

If “B” is zero

The last type of incomplete quadratic equations — when coefficient B is zero. The “middle” of the equation disappears, and we are left only with Ax2Ax^2 alone with some constant term C. These are solved very simply. Ordinary understanding of what a square root is and how to extract it is sufficient.

Quadratic equations when “B” = 0

😀
Elementary
Solve the equation:
2x2−18=02x^2 - 18 = 0

Quadratic equation roots when “B” = 0

Any incomplete quadratic equation with zero coefficient B potentially (if the root can be extracted) has two roots, which can be found by formulas:
Proof
Now you can use the derived general formula to calculate roots instantly:
x2−16=0x1,2=±−−161x1,2=±16x1,2=±4x^2 - 16 = 0 \\ x_{1,2} = \pm \sqrt{-\frac{-16}{1}} \\ x_{1,2} = \pm \sqrt{16} \\ \boxed{x_{1,2} = \pm 4}
3x2+12=0x1,2=±−123x1,2=±−4(no solutions) 3x^2 + 12 = 0 \\ x_{1,2} = \pm \sqrt{-\frac{12}{3}} \\ x_{1,2} = \pm \sqrt{-4} \\ \boxed{\text{(no solutions)}}
5x2−9=0x1,2=±−−95x1,2=±95x1,2=±35 5x^2 - 9 = 0 \\ x_{1,2} = \pm \sqrt{-\frac{-9}{5}} \\ x_{1,2} = \pm \sqrt{\frac{9}{5}} \\ \boxed{x_{1,2} = \pm \frac{3}{\sqrt{5}}}

How to use?

We figured out how to solve all types of incomplete quadratic equations. No cunning tricks or complicated transformations. It was quite simple, right? Moreover, we even derived general formulas for roots for each type of incomplete quadratic equation.

Do not memorize general formulas!

It may seem that since we derived general formulas, they must be memorized. This is not true. Do not learn them. Most people don't even remember them. The most important thing is — to be able to quickly notice that the equation is incomplete (consists of one or two terms), which means it is solved quickly and simply!
Ah, if only quadratic equations in general form could be solved so simply! But things are somewhat more complicated there…
Incomplete Quadratic Equations