The calculator below solves the quadratic equation of
ax2 + bx + c = 0
.
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
ax2 + bx + c = 0
where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.
Derivation of the Quadratic Formula
From this point, it is possible to complete the square using the relationship that:
x2 + bx + c = (x - h)2 + k
Continuing the derivation using this relationship:
Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.
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Quadratic Formula Calculator
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Example: 2x^2-5x-3=0
Step-By-Step Example
Learn step-by-step how to use the quadratic formula!
Example (Click to try)
2x2−5x−3=0
About the quadratic formula
Solve an equation of the form ax2+bx+c=0 by using the quadratic formula:
x=
−b±√b2−4ac
2a
Quadratic Formula Video Lesson
Solve with the Quadratic Formula Step-by-Step [1:29]
Need more problem types? Try MathPapa Algebra Calculator
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Enter your function here. How shall your function be transformed? By in x-direction By in y-direction By to the By to the | |||||
Degree of the function:
( The degree is the highest power of an x. ) Symmetries: y-axis intercept Roots / Maxima / Minima /Inflection points: Characteristic points: Slope at given x-coordinates: |
What are quadratic functions?
Quadratic functions are functions of the form . This means, there is no x to a higher power than . The graph of a quadratic function is a parabola.