October 21st, 2017, 10:34 PM  #1 
Member Joined: May 2015 From: Australia Posts: 77 Thanks: 7  Factoring
A yacht sails a leg of a race by going in a straight course from point A to point B for (x^2 + 8x) km. From point B it travels to point C for (2x+11)km, then back to point A for (4x+24)km. a) write a simplified expression in km for the total distance the yacht sailed b) factorise the expression in (a) c) the factorised expression also represents the product of the length and width of a rectangle, and if the length is 15km, find the value of x. Give your answer as an exact distance d) Find the length of the race, rounded to two decimal places, by substituting the exact value for x into the expression for the total distance in (a) My answer: a) (x^2 + 8x) + (2x+11) + (4x+24) = x^2 + 14x + 35 b) I can't work out how to factorise (x^2 + 14x + 35). Any help would be appreciated. 
October 21st, 2017, 10:53 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,979 Thanks: 1027 
b) you can always use the quadratic formula $\begin{align*} r_1, r_2 &= \dfrac{14 \pm \sqrt{14^2  (4)(35)}}{2} \\ &= \dfrac{14 \pm \sqrt{56}}{2} \\ &= 7 \pm \sqrt{14} \end{align*}$ so $x^2 + 14x + 35 = (xr_1)(xr_2) = (x + 7  \sqrt{14})(x + 7 + \sqrt{14})$ 
October 21st, 2017, 11:10 PM  #3 
Member Joined: May 2015 From: Australia Posts: 77 Thanks: 7 
Thanks, so in this case is there another way to solve it or is the quadratic formula the only way?

October 21st, 2017, 11:20 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 1,979 Thanks: 1027  Quote:
The quadratic formula is the way to go when the factorization isn't immediately apparent.  
October 23rd, 2017, 10:33 AM  #5  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,195 Thanks: 872  Quote:
Here, the quadratic expression is $\displaystyle x^2+ 14x+ 35$. Knowing that $\displaystyle (x+ a)^2= x^2+ 2ax+ a^2$ and comparing those two, 2a= 14 so a= 7, $\displaystyle a^2= 49$. $\displaystyle x^2+ 14x+ 35= x^2+ 14x+ 49 49+ 35= (x+ 7)^2 14= (x+ 7)^2+ (\sqrt{14})^2= (x+ 7+ \sqrt{14})(x+ 7 \sqrt{14})$. Last edited by skipjack; October 24th, 2017 at 05:12 AM.  
October 24th, 2017, 04:02 AM  #6  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,195 Thanks: 872  Quote:
More importantly, "$\displaystyle (x+ 7)^2 14= (x+ 7)^2+ (\sqrt{14})^2$" should have been "$\displaystyle (x+ 7)^2 14= (x+ 7)^2 (\sqrt{14})^2$". Then we can use "$\displaystyle a^2 b^2= (a+ b)(a b)$. Last edited by skipjack; October 24th, 2017 at 05:13 AM.  

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