Question

Bill wants to factor x^2 + 8x + 16 by grouping. however sarah says it is a special product and can be factored a different way. Using complete sentences, explain and demonstrate how both methods will result in the same factors

Answer 1

@Faralen @satellite73

Answer 2

@amistre64

Answer 3

@Orion1213 @Hailey553

Answer 4

what math class are you taking?

Answer 5

Algb 2!

Answer 6

@UsArmy3947 @ranga

Answer 7

omg helllloooo

Answer 8

@Gatorgirl

Answer 9

@jim_thompson5910 @ganeshie8

Answer 10

@thomaster

Answer 11

Factoring by grouping requires GCF or Greatest Common Factor. For a quadratic equation in the form of \[ ax^2 + bx + c\], we can convert it in \[ x^2 +\frac{b}{a}x+\frac{c}{a}\] where the coefficients are related to the roots that determines the factors \[ (x+x_{1})(x+x_{2})=x^2+\frac{b}{a}x+\frac{c}{a}\]\[\frac{b}{a}=x_1+x_2\]\[\frac{c}{a}=(x_1)(x_2)\]

Answer 12

Another method of factoring that we can do for quadratic equation is "completing the square". We start with \[ax^2+bx+c=0\], adding both side by -c, we got \[ax^2+bx+c-c=0-c\]\[ax^2+bx=-c\]dividing both sides by a, we got \[\frac{ax^2+bx}{a}=\frac{-c}{a}\]\[x^2+\frac{b}{a}x=\frac{-c}{a}\]to assure that the left-hand side is a perfect square, we get the half of coefficient of x and then square it, so we have \[\left[ \frac{ 1 }{ 2 }\left( \frac{ b }{ a } \right) \right]^{2}=\frac{ b^2 }{ 4a^2 }\] adding this to both side of the equation, we got \[x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{-c}{a}+\frac{b^2}{4a^2}\]we can now factor completely the left-hand side, while simplifying the right-hand side of the equation, we got \[\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}\]\[\left( x+\frac{ b }{ 2a } \right)=\pm\sqrt{\frac{ b^2-4ac }{ 4a^2 }}=\frac{\pm\sqrt{b^2-4ac}}{2a}\]adding both side by \(-\frac{b}{2a}\), we got \[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]where \[x_1=\frac{-b+\sqrt{b^2-4ac}}{2a}\]\[x_2=\frac{-b-\sqrt{b^2-4ac}}{2a}\]and the factor will be again \[ax^2+bx+c=(x+x_1)(x+x_2)\]

Answer 13

the result \[x=\frac{ -b \pm \sqrt{b^2-4ac} }{ 2a }\]is the popular \(\Large\color{blue}{ Quadratic~Formula...}\)