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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ArticleSummaryPractice

Haste Makes Waste

Method
😀
Elementary
Max needs to solve the following equation at the blackboard:
4x2=80x4x^2 = 80x
He remembered the general formula for solving incomplete quadratic equations when C = 0 and used it:
x1=0x_1 = 0
x2=−BA=−804=−20 x_2 = -\frac{B}{A} = -\frac{80}{4} = -20
Did he solve the equation correctly? Solve the equation, check the roots. If he made a mistake, clearly state what happened and how to avoid it.

Solving incomplete equations

😀
Elementary
First, solve the incomplete quadratic equation manually. After that, be sure to find the roots using the general formulas and compare the results.
x2+x=0x^2 + x = 0

Sign property

Method
😀
Elementary
Prove that if in an incomplete quadratic equation with B = 0 the remaining two coefficients have the same sign, then the equation has no roots. And if the signs are different, then there are two roots.
Using this fact, mentally determine whether the following equations have solutions:
3x2+5=03x^2 + 5 = 0
−2x2=−7 -2x^2 = -7
9999=−x2 9999 = -x^2
−4x2−1+10=0 -4x^2 - 1 + 10 = 0

Stressed Beam

Applied
😀
Elementary
The bending stress σ\sigma in a steel beam supporting a bridge is equal to half the square of its deflection r from the central axis. If the stress exceeds 8, irreversible deformation of the beam will begin. What is the maximum distance from the central axis that a point on the beam can reach without deforming?

Divide, Cannot Pardon

🤔
Intermediate
Masha needed to solve an incomplete quadratic equation with C = 0:
2x2+3x=02x^2 + 3x = 0
She decided to use the same action rule and divided both sides of the equation by x:
2x+3=02x + 3 = 0
Solving this elementary equation, she found the root x=−32x = -\frac{3}{2} and handed the work to the teacher for checking. Did she solve the equation correctly? If not, explain in detail what Masha's mistake is.
Incomplete Quadratic Equations