Given: y=-x^2-3x+2y=−x2−3x+2
Compare to y=ax^2+bx+cy=ax2+bx+c
As the x^2x2 term in negative the general shape is nn∩
color(red)("The y-intercept "=c=+2 ->" point" (x,y)->(0,2))The y-intercept =c=+2→ point(x,y)→(0,2)
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Write in the form y=a(x^2+b/ax)+cy=a(x2+bax)+c
In this case a=-1 and b=-3a=−1andb=−3 giving:
y=-1(x^2+3x)+2y=−1(x2+3x)+2
x_("vertex")=[color(white)(".")(-1/2)xx(b/a)color(white)(".")] -> [color(white)(".")(-1/2)xx(-3)/(-1)color(white)(".")] = -3/2xvertex=[.(−12)×(ba).]→[.(−12)×−3−1.]=−32
So by substitution:
y_("vertex")=-(-3/2)^2-3(-3/2)+2yvertex=−(−32)2−3(−32)+2
y_("vertex")=-9/4+9/2+2 = 4 1/4->17/4yvertex=−94+92+2=414→174
color(red)("Vertex"->(x,y)=(-3/2,17/4))Vertex→(x,y)=(−32,174)
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Notice that (-1)xx(-2)=+2 larr c(−1)×(−2)=+2←c
and that (-1)+(-2) = -3larr b(−1)+(−2)=−3←b
so initially we would think that we have the factorisation. However the negative x^2x2 gives us a problem. Perhaps we can 'force' our initial thoughts to give us the correct form.
Set y=(x-1)(x-2)color(white)("dd") ->color(white)("dd") y=x^2-3x+2 larr" Fail"y=(x−1)(x−2)dd→ddy=x2−3x+2← Fail
Lets try:
y=(-x-1)(x-2)color(white)("dd")->color(white)("dd") y=-x^2+x+2larr" Fail"y=(−x−1)(x−2)dd→ddy=−x2+x+2← Fail
Ok! Looks as though we do not have whole number factorisation. So lets use the formula x=(-b+-sqrt(b^2-4ac))/(2a)x=−b±√b2−4ac2a
y=color(white)("dd.d")ax^2+bx+cy=dd.dax2+bx+c
y=(-1)x^2-3x+2y=(−1)x2−3x+2
So a=-1; b=-3 and c=+2a=−1;b=−3andc=+2 giving:
x=(+3+-sqrt((-3)^2-4(-1)(+2)))/(2(-1))x=+3±√(−3)2−4(−1)(+2)2(−1)
x=-3/2+-sqrt(17)/2 larr" Exact solution"x=−32±√172← Exact solution
As 17 is a prime number we can not simplify this any further.
Approximate solution:
x=-1.5+-2.062x=−1.5±2.062 to 3 decimal places giving:
color(red)(x_("intercept")~~ -3.562 and +0.562" to 3 decimal places")xintercept≈−3.562and+0.562 to 3 decimal places
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