In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions. Given a general quadratic equation of the form a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} , with x {\displaystyle x} representing an unknown, and coefficients a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} representing known real or complex numbers with a ≠ 0 {\displaystyle a\neq 0} , the values of x {\displaystyle x} satisfying the equation, called the roots or zeros, can be found using the quadratic formula,
x = − b ± b 2 − 4 a c 2 a , {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},}
where the plus–minus symbol " ± {\displaystyle \pm } " indicates that the equation has two roots. Written separately, these are:
x 1 = − b + b 2 − 4 a c 2 a , x 2 = − b − b 2 − 4 a c 2 a . {\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}},\qquad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}.}
The quantity Δ = b 2 − 4 a c {\displaystyle \textstyle \Delta =b^{2}-4ac} is known as the discriminant of the quadratic equation. If the coefficients a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are real numbers then when Δ > 0 {\displaystyle \Delta >0} , the equation has two distinct real roots; when Δ = 0 {\displaystyle \Delta =0} , the equation has one repeated real root; and when Δ < 0 {\displaystyle \Delta <0} , the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other. Geometrically, the roots represent the x {\displaystyle x} values at which the graph of the quadratic function y = a x 2 + b x + c {\displaystyle \textstyle y=ax^{2}+bx+c} , a parabola, crosses the x {\displaystyle x} -axis: the graph's x {\displaystyle x} -intercepts. The quadratic formula can also be used to identify the parabola's axis of symmetry.
Derivation by completing the square
The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle \textstyle ax^{2}+bx+c=0} . The idea is to transform the equation into the form ( x + k ) 2 = s {\displaystyle \textstyle (x+k)^{2}=s} for some expressions k {\displaystyle k} and s {\displaystyle s} written in terms of the coefficients; take the square root of both sides; and then isolate x {\displaystyle x} . We start by dividing the equation by the quadratic coefficient a {\displaystyle a} , which is allowed because a {\displaystyle a} is non-zero. Afterwards, we subtract the constant term c / a {\displaystyle c/a} to isolate it on the right-hand side:
… excerpt ends here. Continue reading the full article.
