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ax2 + bx + c = 0 a = b = c = Lesson on the Quadratic FormulaLesson ContentsWhat is the Quadratic Formula?The quadratic formula is a formula that is used to solve a quadratic equation of the standard form ax2 + bx + c = 0. The quadratic formula is given as: $$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$$ Where x denotes the solution(s) to the quadratic equation, and a, b, and c are the coefficients of the quadratic equation. The part inside the square root (b2 – 4ac) is called the discriminant and it tells us how many roots the quadratic equation has. How to Solve a QuadraticTo solve a quadratic equation, we must make sure it is in standard form. As shown earlier, standard form for a quadratic is ax2 + bx + c = 0. Once in standard form, we plug the equation’s coefficients into the quadratic formula. If the discriminant is a positive number, continue solving until the two real solutions are obtained. Sometimes the discriminant will not be positive which results in us either taking the square root of zero or a negative number. If the discriminant is zero, continue solving the quadratic formula as usual until the final, single solution is obtained. That final number is the solution, but we call it a repeated real solution. This means the graph of the quadratic touches the x-axis twice in the same spot. If the discriminant is a negative number, taking the square root of it results in an imaginary number. Therefore, the quadratic equation has two imaginary solutions and no real solutions. How the Calculator WorksThe quadratic formula solver on this page is written in JavaScript (JS) and is powered by a JS native computer algebra system (CAS). The calculations all occur within your device’s internet browser JS engine, which allows for near-instantaneous answers and no waiting on the page to communicate with a server or refresh. When the calculate button is pressed, your coefficients are fed to the CAS which then uses quadratic solving steps to symbolically calculate the solution(s). Being symbolic in nature, the CAS treats the quadratic like a human would with paper and pencil and maintains near-perfect accuracy. When the CAS is done solving, the answer is converted to LaTeX (a math rendering language) and displayed in the answer area. If an error occurs or the inputs do not allow for a solution to be calculated, an error notice will instead be displayed.
Solve equations and systems of equations with Wolfram|AlphaA powerful tool for finding solutions to systems of equations and constraintsWolfram|Alpha is capable of solving a wide variety of systems of equations. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Additionally, it can solve systems involving inequalities and more general constraints. 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 solving systems of equations.
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What are systems of equations?A system of equations is a set of one or more equations involving a number of variables.The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection. Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent). More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers. Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science. How do you know how many solutions a quadratic equation has?Solution:. if the discriminant is positive. , then the quadratic equation has two solutions.. if the discriminant is equal. , then the quadratic equation has one solution.. if the discriminant is negative. , then the quadratic equation has no solution.. How do you know how many solutions are in a parabola?If the parabola and the line touch at a single point, then there is one solution that is true for both equations. If the graphs of the equations do not intersect, then there are no solutions that are true for both equations.
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Number of solutions of quadratic equations calculator
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