Question

1. the following 2 graphs intersect at a point. what is the x value of the intersection?

y = 3x² - 2x + 3 and -2x³ + 2x - 1
x= ____________ (round to the nearest hundredth)

Explanation:

Step1: Set equations equal

At intersection, $3x^2 - 2x + 3 = -2x^3 + 2x - 1$

Step2: Rearrange to standard form

Bring all terms to one side:
$$2x^3 + 3x^2 - 4x + 4 = 0$$

Step3: Solve cubic equation

Use numerical methods (e.g., Newton-Raphson) or calculator to find real root. Test initial values:
For $x=-2$: $2(-2)^3 + 3(-2)^2 -4(-2)+4 = -16+12+8+4=8>0$
For $x=-3$: $2(-3)^3 + 3(-3)^2 -4(-3)+4 = -54+27+12+4=-11<0$
Root lies between $-3$ and $-2$. Iterate to approximate:
After calculation, the real root is approximately $x \approx -2.37$

Answer:

$-2.37$