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The solutions of the quadratic equation ax 2 + bx + c = 0 correspond to the roots of the function f(x) = ax 2 + bx + c, since they are the values of x for which f(x) = 0. If a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis.
The quadratic formula is exactly correct when performed using the idealized arithmetic of real numbers, but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available on computers, the limitations of the number representation ...
where x is the variable, and a, b, and c represent the coefficients. Such polynomials often arise in a quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.} The solutions to this equation are called the roots and can be expressed in terms of the coefficients as the quadratic formula .
Integrals involving R = √ ax 2 + bx + c. Assume (ax 2 + bx + c) cannot be reduced to the following expression (px + q) 2 for some p and q.
Quadratic equation, a polynomial equation of degree 2 (reducible to 0 = ax 2 + bx + c) Quadratic formula, calculation to solve a quadratic equation for the independent variable (x) Quadratic field, an algebraic number field of degree two over the field of rational numbers
Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies a = 0 or b = 0 or c = 0. If the original fact were stated as "ab = 0 implies a = 0 or b = 0", then when saying "consider abc = 0," we would have a conflict of terms when ...
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This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic. The formula can be proved as follows: Starting from the equation t 3 + pt + q = 0 , let us set t = u cos θ .