Quadratic equations have three terms in the standard form and were first used by Babylonian mathematicians. They can be solved using factoring, completing the square, or the quadratic formula, with the discriminant indicating the type and number of solutions. Online solvers are available.
A quadratic equation consists of a single variable with three terms in the standard form: ax2 + bx + c = 0. The first quadratic equations were developed as a method used by Babylonian mathematicians around 2000 BC to solve simultaneous equations. Quadratic equations can be applied to physics problems involving parabolic motion, path, shape, and stability. Several methods have evolved to simplify the solution of such equations for the variable x. You can find any number of quadratic equation solvers online, where you can enter and automatically calculate the values of the coefficients of your quadratic equation.
The three most commonly used methods for solving quadratic equations are factoring, completing the square, and the quadratic formula. Factoring is the simplest form of solving a quadratic equation. When the quadratic equation is in its standard form, it is easy to see whether the constants a, b and c are such that the equation represents a perfect square. First, the standard modulus must be divided by a. Then, half of what is now, the b/a term must equal twice, what is now, the c/a term; if this is true, then the standard form can be decomposed into the perfect square of (x ± d)2.
If the solution to a quadratic equation is not a perfect square and the equation cannot be factored in its current form, a second solution method, completing the square, can be used. After dividing by the a term, the b/a term is divided by two, squared, and then added to both sides of the equation. The square root of the perfect square can be equaled to the square root of all remaining constants on the right side of the equation to find x.
The final method for solving the standard quadratic equation is to directly substitute the constant coefficients (a, b and c) into the quadratic formula: x = (-b±sqrt(b2-4ac))/2a, which was derived from the completion method of the squares in the generalized equation. The discriminant of the quadratic formula (b2 – 4ac) appears under the square root sign and, even before the equation is solved for x, it can indicate the type and number of solutions found. The type of solution depends on whether the discriminant is equal to the square root of a positive or negative number. When the discriminant is zero, there is only one positive root. When the discriminant is positive, there are two positive roots and when the discriminant is negative, there are both positive and negative roots.
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