When deriving the quadratic formula by completing the square, what expression can be added to both sides of the equation to create a perfect square trinomial?
Answers
Answer:
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Answer:
general quadratic equation is:
ax^2+bx+c=0ax2+bx+c=0
Moving the constant term to the other side of the equation, we get:
ax^{2}+bx=-cax2+bx=−c
Dividing both sides by a, we get:
x^{2}+\frac{b}{a}(x)=-\frac{c}{a}x2+ab(x)=−ac
This is the equation we have been given in the question.
The general equation for a perfect square trinomial is:
(x+y)^{2}=x^{2}+2xy+y^{2}(x+y)2=x2+2xy+y2
Re-writing the given equation we get:
x^{2}+2(\frac{b}{2a})(x)=-\frac{c}{a}x2+2(2ab)(x)=−ac
Comparing this equation to the general square trinomial, we can write that we have:
Square of first term i.e. x^{2}x2
Twice the product of first and second term i.e. 2(\frac{b}{2a})(x)2(2ab)(x)
So in order to complete the square we need to add the square of second term to the both sides of the equation. From the product we can see that the second term is \frac{b}{2a}2ab
So, the square of \frac{b}{2a}2ab should be added to both sides to get a perfect square trinomial.
\begin{gathered}x^{2}+2(\frac{b}{2a})(x)+(\frac{b}{2a} )^{2}=-\frac{c}{a}+(\frac{b}{2a} )^{2}\\\\(x+\frac{b}{2a})^{2}=-\frac{c}{a}+(\frac{b}{2a} )^{2}\end{gathered}x2+2(2ab)(x)+(2ab)2=−ac+(2ab)2(x+2ab)2=−ac+(2ab)2
Thus, the answer in both blanks would be: (\frac{b}{2a} )^{2}=\frac{b^{2}}{4a^{2}}(2ab)2=4a2b2