Completing the square: Difference between revisions
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In [[algebra]], '''completing the square''' is a way of rewriting a quadratic [[polynomial]] as the sum of a constant and a constant multiple of the square of a first-degree polynomial. Thus one has | In [[algebra]], '''completing the square''' is a way of rewriting a quadratic [[polynomial]] as the sum of a constant and a constant multiple of the square of a first-degree polynomial. Thus one has | ||
:<math> ax^2 + bx + c = a( | :<math> ax^2 + bx + c = a(\cdots)^2 + \text{constant} </math> | ||
and completing the square is the way of filling in the blank between the brackets. Completing the square is used for solving quadratic | and completing the square is the way of filling in the blank between the brackets. Completing the square is used for solving [[quadratic equation]]s (the proof of the well-known formula for the general solution consists of completing the square). The technique is also used to find the maximum or minimum value of a quadratic function, or in other words, the vertex of a [[parabola]]. | ||
The technique relies on the [[elementary algebra|elementary algebraic]] identity | The technique relies on the [[elementary algebra|elementary algebraic]] identity | ||
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</math> | </math> | ||
Now the expression ( | Now the expression (<math> \scriptstyle x^2 + 2 x \cdot 7</math>) corresponds to <math> \scriptstyle u^2 + 2 u v</math> in the elementary identity labeled (*) above. If <math> \scriptstyle x^2 </math> is <math> \scriptstyle u^2 </math> and <math> \scriptstyle 2 x \cdot 7 </math> is <math> \scriptstyle 2 u v </math>, then <math> \scriptstyle v </math> must be 7. Therefore <math> \scriptstyle ( u + v )^2 </math> must be <math> \scriptstyle ( x + 7 )^2 </math>. So we continue: | ||
: <math> | : <math> | ||
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</math> | </math> | ||
Now we have added | Now we have added <math> \scriptstyle 7^2 </math> ''inside'' the parentheses, and compensated (thus justifying the "=") by subtracting <math> \scriptstyle 3 (7^2) </math> ''outside'' the parentheses. The expression ''inside'' the parentheses is now <math> \scriptstyle u^2 + 2 u v + v^2 </math>, and by the elementary identity labeled (*) above, it is therefore equal to <math> \scriptstyle ( u + v )^2 </math>, i.e. to <math> \scriptstyle ( x + 7 )^2 </math>. So now we have | ||
: <math> 3(x + 7)^2 - 5 - 3(7^2) = 3(x + 7)^2 - 152.\, </math> | : <math> 3(x + 7)^2 - 5 - 3(7^2) = 3(x + 7)^2 - 152.\, </math> | ||
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It is possible to give a closed formula for the completion in terms of the coefficients ''a'', ''b'' and ''c''. Namely, | It is possible to give a closed formula for the completion in terms of the coefficients ''a'', ''b'' and ''c''. Namely, | ||
: <math> ax^2+bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a}, </math> | : <math> ax^2+bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a}, </math> | ||
where <math>\Delta</math> stands for the well-known ''discriminant'' of the polynomial, that is <math>\Delta=b^2-4ac | where <math> \scriptstyle \Delta </math> stands for the well-known ''discriminant'' of the polynomial, that is <math> \scriptstyle \Delta = b^2 - 4ac </math>. | ||
Indeed, we have | Indeed, we have | ||
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</math> | </math> | ||
The last expression inside parentheses above corresponds to | The last expression inside parentheses above corresponds to <math> \scriptstyle u^2 + 2 u v </math> in the identity labelled (*) above. We need to add the third term, <math> \scriptstyle v^2 </math>: | ||
: <math> | : <math> | ||
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</math> | </math> | ||
<!--[''To be continued...'']--> | <!--[''To be continued...'']-->[[Category:Suggestion Bot Tag]] |
Latest revision as of 11:01, 31 July 2024
In algebra, completing the square is a way of rewriting a quadratic polynomial as the sum of a constant and a constant multiple of the square of a first-degree polynomial. Thus one has
and completing the square is the way of filling in the blank between the brackets. Completing the square is used for solving quadratic equations (the proof of the well-known formula for the general solution consists of completing the square). The technique is also used to find the maximum or minimum value of a quadratic function, or in other words, the vertex of a parabola.
The technique relies on the elementary algebraic identity
Concrete examples
We want to fill in this blank:
We write
Now the expression () corresponds to in the elementary identity labeled (*) above. If is and is , then must be 7. Therefore must be . So we continue:
Now we have added inside the parentheses, and compensated (thus justifying the "=") by subtracting outside the parentheses. The expression inside the parentheses is now , and by the elementary identity labeled (*) above, it is therefore equal to , i.e. to . So now we have
Thus we have the equality
More abstractly
It is possible to give a closed formula for the completion in terms of the coefficients a, b and c. Namely,
where stands for the well-known discriminant of the polynomial, that is .
Indeed, we have
The last expression inside parentheses above corresponds to in the identity labelled (*) above. We need to add the third term, :