How to Solve a Quadratic Equation with Only Two Terms
A quadratic equation with only two terms is a simplified form of the standard quadratic equation $ ax^2 + bx + c = 0 $, where either the linear term ($ bx $) or the constant term ($ c $) is missing. As long as $ a \neq 0 $ and the $ x^2 $ term remains, the equation is still quadratic. These equations are simpler to solve than full three-term versions because they allow direct methods like factoring or taking square roots.
There are two main types of two-term quadratics:
- $ ax^2 + bx = 0 $ — missing constant term
- $ ax^2 + c = 0 $ — missing linear term
Each type follows specific solving patterns based on algebraic structure.
Solving $ ax^2 + bx = 0 $: Factor Out $ x $
When the constant term is missing, both terms share a common factor of $ x $. Factoring it out gives:
$$ x(ax + b) = 0 $$
Using the zero-product property—where a product equals zero only if at least one factor is zero—we set each factor equal to zero:
- $ x = 0 $
- $ ax + b = 0 $ → $ x = -\frac{b}{a} $
This yields two solutions: $ x = 0 $ and $ x = -\frac{b}{a} $.
Example: $ 2x^2 - 8x = 0 $
Factor out $ 2x $:
$$ 2x(x - 4) = 0 $$
Set each factor to zero:
- $ 2x = 0 $ → $ x = 0 $
- $ x - 4 = 0 $ → $ x = 4 $
The solutions are $ x = 0 $ and $ x = 4 $. This method is efficient and avoids unnecessary steps like using the quadratic formula.
A common mistake is dividing both sides by $ x $ early, such as turning $ x^2 = 3x $ into $ x = 3 $. This omits the solution $ x = 0 $. Always move all terms to one side first and factor.
Solving $ ax^2 + c = 0 $: Isolate $ x^2 $ and Take Square Roots
When the linear term is missing, isolate $ x^2 $:
$$ ax^2 = -c \quad \Rightarrow \quad x^2 = -\frac{c}{a} \quad \Rightarrow \quad x = \pm \sqrt{-\frac{c}{a}} $$
The $ \pm $ indicates two solutions: one positive, one negative. However, the nature of the solutions depends on the sign of $ -\frac{c}{a} $.
- If $ -\frac{c}{a} > 0 $, the result under the square root is positive → two real solutions
- If $ -\frac{c}{a} < 0 $, the result is negative → two imaginary solutions involving $ i = \sqrt{-1} $
For example:
- $ x^2 - 9 = 0 $ → $ x^2 = 9 $ → $ x = \pm 3 $ (real)
- $ x^2 + 4 = 0 $ → $ x^2 = -4 $ → $ x = \pm 2i $ (imaginary)
The sign of $ a $ does not affect whether solutions exist. In $ -x^2 + 16 = 0 $, rearranging gives $ x^2 = 16 $, so $ x = \pm 4 $. The negative coefficient only affects the parabola’s direction when graphed.
An edge case is $ 5x^2 = 0 $, which has a repeated root: $ x = 0 $ twice. This is called a double root and counts as two identical solutions in algebra.
The Quadratic Formula Still Applies
The general quadratic formula is:
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
It works for two-term equations, but simplifies significantly when one term is missing.
If $ c = 0 $, the formula becomes:
$$ x = \frac{-b \pm \sqrt{b^2}}{2a} = \frac{-b \pm |b|}{2a} $$
This gives $ x = 0 $ and $ x = -\frac{b}{a} $, matching the factoring method.
If $ b = 0 $, it reduces to:
$$ x = \frac{\pm \sqrt{-4ac}}{2a} = \pm \sqrt{-\frac{c}{a}} $$
which matches the square root method.
While the formula is reliable, it’s often slower than direct approaches for two-term cases. It’s best used for verification or when the method isn’t immediately clear.
Key Takeaways
- A quadratic equation with only two terms lacks either the linear term ($ bx $) or the constant term ($ c $), but still has $ x^2 $ with $ a \neq 0 $.
- If the equation is $ ax^2 + bx = 0 $, factor out $ x $ and apply the zero-product property: solutions are $ x = 0 $ and $ x = -\frac{b}{a} $.
- If the equation is $ ax^2 + c = 0 $, isolate $ x^2 $ and take the square root: $ x = \pm \sqrt{-\frac{c}{a}} $; solutions may be real or imaginary depending on the sign of $ -\frac{c}{a} $.
- Avoid dividing by the variable early—this can eliminate the $ x = 0 $ solution. Always factor instead.
- The quadratic formula works but is usually unnecessary; simpler methods are faster and clearer for two-term cases.
- Equations like $ 5x^2 = 0 $ have a double root at $ x = 0 $.
- Two-term quadratics appear in real-world contexts, such as physics (e.g., free-fall motion $ h_0 = \frac{1}{2}gt^2 $) and geometry (e.g., $ s^2 = A $ for square area), where one variable depends quadratically on another Wikipedia — Quadratic equation, Khan Academy.
