Find the QUADRATIc equation 100 POINTS PLS HELP (1,6),(2,19),(3,38),(4,63),(5,94)

Step-by-step explanation:

we are given some coordinates

(1,6),(2,19),(3,38),(4,63),(5,94)

we want to figure out the quadratic equation which passes those coordinates

notice that, the coordinates in quadratic sequence

we know that,

A sequence is quadratic if and only if it has second difference

recall that,

these formulas are necessary to figure out a,b and c

because we know that

Quadratic equation standard form:

therefore to figure out a,b and c these formulas are important

let's figure out a:

we got that 2nd difference is 6 thus

divide both sides by 2

let's figure out b:

by using the second formula we can figure out b

our and are 6 and 19 respectively

substitute:

simplify multiplication:

simplify substraction:

cancel 9 from both sides:

let's figure out c:

our is 6

substitute:

simplify addition:

cancel 7 from both sides:

altogether substitute a,b and c to quadratic equation:

and we are done!

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4

for the free

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