Quadratic formulas and problems

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All the formulas, concepts, and methods for solving quadratic equations, with the logic behind their derivation. A set of mixed problems that reduce to quadratic trinomials or quadratic equations.

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Summary
Practice
In the problems on this page, you can use all methods and formulas related to quadratic equations:
  • 1
    Incomplete form
  • 2
    Completing the square
  • 3
    Quadratic formula
  • 4
    Factoring
  • 5
    Vieta’s Formulas
  • 6
    Solving quadratic equations mentally
    • Throw everything into the fight in any order!
    The main thing is to come out on top in the most spectacular and efficient way. 😎

    This is the finale, not the beginning!

    The problems here assume that you already know how to use the formulas! If this is a problem for you, make sure to work through the practice pages from each of the topics listed above, and then come back here.

    A Number and Its Reciprocal

    😀
    Elementary
    The sum of a number and its reciprocal is 52\dfrac{5}{2}. Find the number.

    Consecutive Squares

    😀
    Elementary
    The sum of the squares of two consecutive even numbers is 244. Find these numbers.

    Everything at Once!

    😀
    Elementary
    Solve the quadratic equation in every possible way, and then factor it:
    4m2+m−3=04m^2 + m - 3 = 0

    Reading the Coffee Grounds

    Method
    🤔
    Intermediate
    Solve the quadratic equations without using the quadratic formula:
    463x2−102x−361=0463x^2 - 102x - 361 = 0

    Shadowboxing

    Elegant
    🤔
    Intermediate
    Solve the quadratic equation without using the quadratic formula:
    2x2−5x−7=2⋅(35)2−5⋅(35)−72x^2 - 5x - 7 = 2 \cdot \left(\frac{3}{5}\right)^2 - 5 \cdot \left(\frac{3}{5}\right) - 7

    Slow Train

    🤔
    Intermediate
    A train travels 200 kilometers at a constant speed. If it were moving 10 kilometers per hour faster, it would cover this distance 1 hour sooner. At what speed was the train traveling?

    Fine Tuning

    🤔
    Intermediate
    For which values of k is the product of the roots of the quadratic equation below equal to zero?
    x2+3x+(k2−7k+12)=0x^2 + 3x + (k^2 - 7k + 12) = 0

    The Supremacy of “C”

    🤔
    Intermediate
    Derive general formulas for solving quadratic equations of the form:
    x2+2x+C=0x^2 + 2x + C = 0
    Under what conditions does this equation have solutions?

    Eternal Optimist

    🤔
    Intermediate
    Is it true that if all terms of a quadratic equation in general form have a + sign, for example 3x2+x+100=03x^2 + x + 100 = 0, then it has no solutions? After all, we are adding three positive numbers, and there is no way their sum can be 0!
    If this is not true, then give a counterexample and explain in detail why this logic is wrong.

    Variable Substitution

    Method
    Elegant
    🤔
    Intermediate
    Solve the equation:
    x4−7x2−144=0x^4 - 7x^2 - 144 = 0

    Blood and Sweat

    🤯
    Advanced
    Solve the equation:
    x2+x+23x2+5x−14=x2+x+63x2+5x−10\frac{x^2+x+2}{3x^2+5x-14} = \frac{x^2+x+6}{3x^2+5x-10}

    Ups and Downs

    Method
    🤯
    Advanced
    For what value of the parameter m is the sum of the squares of the roots of the equation smallest?
    x2+(2−m)x−m−3=0x^2 + (2-m)x - m - 3 = 0

    Quadratic Root Classes

    Elegant
    🤯
    Advanced
    The three equations below have the same roots and differ from each other only by a common factor by which all coefficients were multiplied.
    x2+6x−7=0Monic\underset{\text{Monic}}{x^2 + 6x - 7 = 0}
    2x2+12x−14=0Multiplied by 2 \underset{\text{Multiplied by }2}{2x^2 + 12x - 14 = 0}
    3x2+18x−21=0Multiplied by 3 \underset{\text{Multiplied by }3}{3x^2 + 18x - 21 = 0}
    By dividing both sides of the second equation by 2, and the third by 3, we get the original first equation again. By multiplying both sides of the second equation by 32\frac{3}{2}, we get the third equation. In short, all three equations can be reduced to each other by multiplying by some number.
    But do there exist two equations with the same roots that cannot be reduced to each other by multiplication by some number? If they do exist, give an example of such equations. If they do not, prove it rigorously.
    Quadratic formulas and problems