Question

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

Answer 1

Answer:

$\displaystyle 3 {x}^{2} + 4x - 1 = 0$

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,

• $\displaystyle 2a=2^{ \text{nd }} \text{diff}$
• $\displaystyle 3a + b = U_{2}-U_1$
• $\displaystyle a + b + c = U_1$

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

because we know that

Quadratic equation standard form:

$\displaystyle a {x}^{2} + bx + c = 0$

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

$\displaystyle 2a = 6$

divide both sides by 2

$\displaystyle \frac{2a}{2} =\frac{ 6}{2}$

$\displaystyle a = 3$

let's figure out b:

by using the second formula we can figure out b

our $U_1$ and $U_2$ are 6 and 19 respectively

substitute:

$3 \cdot 3 + b = 19 - 6$

simplify multiplication:

$9+ b = 19 - 6$

simplify substraction:

$9+ b = 13$

cancel 9 from both sides:

$b = 4$

let's figure out c:

our $U_1$ is 6

substitute:

$\displaystyle 3 + 4 + c = 6$

simplify addition:

$\displaystyle 7 + c = 6$

cancel 7 from both sides:

$\displaystyle c = - 1$

altogether substitute a,b and c to quadratic equation:

$\displaystyle 3 {x}^{2} + 4x - 1 = 0$

and we are done!