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The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation [ 3 ] a x 2 + b x + c = a ( x − r ) ( x − s ) = 0 {\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0} where r and s are the solutions for x .
Denoting the two roots by r 1 and r 2 we distinguish three cases. If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is not zero, and |r 1 | ≠ |r 2 |, the continued fraction converges to the root of maximum modulus (i.e., to the root with the greater absolute value).
Finding roots of 3x 2 + 5x − 2. Lill's method can be used with Thales's theorem to find the real roots of a quadratic polynomial. In this example with 3x 2 + 5x − 2, the polynomial's line segments are first drawn in black, as above. A circle is drawn with the straight line segment joining the start and end points forming a diameter.
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
This quadratic equation has two solutions: = and = But if 0 {\displaystyle 0} is substituted for x {\displaystyle x} in the original equation, the result is the invalid equation 2 = 0 {\displaystyle 2=0} .
In particular, the two solutions of a quadratic equation are conjugate, as per the in the quadratic formula =. Complex conjugation is the special case where the square root is i = − 1 , {\displaystyle i={\sqrt {-1}},} the imaginary unit .