We'll do this last last one real slow.
[tex]x^2 + 10 x + 25 = 8[/tex]
Step one, move the 8 to the left side by subtracting it from 25:
[tex]x^2 + 10x + 17 = 0[/tex]
To factor we'd need factors of 17 which add to 10. The only factors of 17 are 17 and 1 so there's no factoring. We use the quadratic formula.
[tex]x= \frac 1 {2a}(-b \pm \sqrt{b^2-4ac}) [/tex]
a, b and c refer to the coefficients on the quadratic equation. We have [tex]x^2 + 10 x + 17 = 0[/tex] so a=1 (the coefficient on x^2) b=10, c=17.
[tex] x= \frac 1 2(-10 \pm \sqrt{10^2-4(17)})[/tex]
Now we simplify the square root. 10²=100, 4(17)=68, so
[tex]x = \frac 1 2(-10 \pm \sqrt{32}) [/tex]
32 = 16(2) so we can simplify a bit more.
[tex]x = \frac 1 2(-10 \pm 4 \sqrt{2})[/tex]
[tex]x = -5 \pm 2 \sqrt{2}[/tex]
We compare to the answers and get
Answers: A E
With the Shakespeare Quadratic Formula shortcut
[tex]x^2-2bx+c \textrm{ has zeros } x=b \pm \sqrt{b^2-c}[/tex]
Using that on [tex]x^2 + 10x + 17 = 0[/tex] we could have gotten here a bit quicker. b=-5 this time, c=17, so
[tex]x=-5 \pm \sqrt{25-17} = -5 \pm \sqrt 8 = -5 \pm 2 \sqrt 2[/tex]
Good luck. Please ask questions if you're still unsure.
