The calculator below solves the quadratic equation of Show
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 FormulaFrom 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. We can help you solve an equation of the form "ax2 + bx + c = 0" algebra/images/quadratic-solver.js Is it Quadratic?Only if it can be put in the form ax2 + bx + c = 0, and a is not zero. The name comes from "quad" meaning square, as the variable is squared (in other words x2). These are all quadratic equations in disguise:
How Does this Work?The solution(s) to a quadratic equation can be calculated using the Quadratic Formula: The "±" means we need to do a plus AND a minus, so there are normally TWO solutions ! The blue part (b2 - 4ac) is called the "discriminant", because it can "discriminate" between the possible types of answer:
Learn more at Quadratic Equations Note: you can still access the old version here.
Apply the quadratic formula using Wolfram|AlphaA useful tool for finding the solutions to quadratic equationsWolfram|Alpha can apply the quadratic formula to solve equations coercible into the form . In doing so, Wolfram|Alpha finds both the real and complex roots of these equations. It can also utilize other methods helpful to solving quadratic equations, such as completing the square, factoring and graphing. Learn more about:
Tips for entering queriesEnter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about finding roots of quadratic equations.
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What are quadratic equations, and what is the quadratic formula?A quadratic is a polynomial of degree two.Quadratic equations form parabolas when graphed, and have a wide variety of applications across many disciplines. In physics, for example, they are used to model the trajectory of masses falling with the acceleration due to gravity. Situations arise frequently in algebra when it is necessary to find the values at which a quadratic is zero. In other words, it is necessary to find the zeros or roots of a quadratic, or the solutions to the quadratic equation. Relating to the example of physics, these zeros, or roots, are the points at which a thrown ball departs from and returns to ground level. One common method of solving quadratic equations involves expanding the equation into the form and substituting the , and coefficients into a formula known as the quadratic formula. This formula, , determines the one or two solutions to any given quadratic. Sometimes, one or both solutions will be complex valued. Discovered in ancient times, the quadratic formula has accumulated various derivations, proofs and intuitions explaining it over the years since its conception. Some involve geometric approaches. Others involve analysis of extrema. There are also many others. Those listed and more are often topics of study for students learning the process of solving quadratic equations and finding roots of equations in general. Alternative methods for solving quadratic equations do exist. Completing the square, factoring and graphing are some of many, and they have use cases—but because the quadratic formula is a generally fast and dependable means of solving quadratic equations, it is frequently chosen over the other methods. |
Find the zeros of a quadratic equation calculator
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