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There is no quadratic equation that is 'linear'. There are linear equations and quadratic equations. Linear equations are equations in which the degree of the variable is 1, and quadratic equations are those equations in which the degree of the variable is 2.
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There are many ways quadratic equations are used in the real world. These equations are used to calculate area, speed and profit
The quadratic formula is used to solve the quadratic equation. Many equations in which the variable is squared can be written as a quadratic equation, and then solved with the quadratic formula.
Because it's part of the quadratic equation formula in finding the roots of a quadratic equation.