In analytical geometry , this method is one of the methods with which equations can be converted from quadrics to a normal form. In doing so, quadratic terms are transformed into several variables ( quadratic forms ).
${\ displaystyle y = a \ left (x ^ {2} + {\ frac {b} {a}} x \ right) + c}$
The parenthesized term is now brought into a form so that the first binomial formula can be used. It is referred to as “nutritious zero”, or “zero supplement”.
${\ displaystyle (x ^ {2} + 2dx + d ^ {2}) - d ^ {2}}$${\ displaystyle d ^ {2} -d ^ {2}}$
Square addition:
${\ displaystyle y = a \ left (x ^ {2} + {\ frac {b} {a}} x + \ left ({\ frac {b} {2a}} \ right) ^ {2} - \ left ( {\ frac {b} {2a}} \ right) ^ {2} \ right) + c}$
Formation of the square:
${\ displaystyle y = a \ left [\ left (x + {\ frac {b} {2a}} \ right) ^ {2} - \ left ({\ frac {b} {2a}} \ right) ^ {2 } \ right] + c}$
${\ displaystyle y = a \ left (x + {\ frac {b} {2a}} \ right) ^ {2} - {\ frac {ab ^ {2}} {4a ^ {2}}} + c}$
Vertex form of the function:
${\ displaystyle y = a \ left (x + {\ frac {b} {2a}} \ right) ^ {2} + \ left (c - {\ frac {b ^ {2}} {4a}} \ right) }$
Reading the vertex:
${\ displaystyle S \ left (- {\ frac {b} {2a}} \ right | \ left.c - {\ frac {b ^ {2}} {4a}} \ right)}$
Addition: With is the coordinate of the vertex. The following then applies to the associated coordinate .
${\ displaystyle x_ {S} = - b / (2a)}$${\ displaystyle x_ {S}}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle y_ {S}}$${\ displaystyle y_ {S} = ca \ cdot (x_ {S}) ^ {2}}$
example
Given quadratic function:
${\ displaystyle y = 2x ^ {2} -12x + 13 \,}$
Excluding the leading coefficient:
${\ displaystyle y = 2 (x ^ {2} -6x) +13 \,}$
Because of the "nutritious zero" is inserted:
${\ displaystyle ({\ tfrac {6} {2}}) ^ {2} = 9}$${\ displaystyle 9-9}$
The left side of the equation is now shaped so that the second binomial formula can be applied. is also added on the right side of the equation:
${\ displaystyle x ^ {2} -2dx + d ^ {2}}$${\ displaystyle d ^ {2}}$
should be brought to affine normal form. Adding a square in the variable (i.e. , considered a parameter) and then adding a square in gives
${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle y}$
With the substitution , the equation of the quadric is transformed to the circle equation .
${\ displaystyle u = x + 2y-3}$${\ displaystyle v = y-1}$${\ displaystyle Q}$${\ displaystyle u ^ {2} + v ^ {2} = 1}$
Alternatives
The vertex form of a quadratic function can also be obtained with the help of differential calculus (by determining the zero of the first derivative).
For solving quadratic equations, there are ready-made solution formulas that you only have to insert. The derivation of these formulas is done using the quadratic addition.
literature
FA Willers, KG Krapf: Elementary Mathematics: A preliminary course for higher mathematics . 14th edition. Springer, 2013, ISBN 978-3-642-86564-0 , pp. 84–86