Question

deux fonctions dont les règles sont :
y = 0,3(x - 6,1)² + 5,3
y = 0,6x + 2,54
sont représentées dans le plan cartésien ci-dessous.

Explanation:

To find the intersection points of the two functions \( y = 0.3(x - 6.1)^2 + 5.3 \) and \( y = 0.6x + 2.54 \), we set them equal to each other:

Step 1: Set the equations equal

$$ 0.3(x - 6.1)^2 + 5.3 = 0.6x + 2.54 $$

Step 2: Expand the square

First, expand \( (x - 6.1)^2 \):

$$ (x - 6.1)^2 = x^2 - 12.2x + 37.21 $$

Substitute back into the equation:

$$ 0.3(x^2 - 12.2x + 37.21) + 5.3 = 0.6x + 2.54 $$

Step 3: Distribute the 0.3

$$ 0.3x^2 - 3.66x + 11.163 + 5.3 = 0.6x + 2.54 $$

Step 4: Combine like terms

$$ 0.3x^2 - 3.66x + 16.463 = 0.6x + 2.54 $$

Subtract \( 0.6x \) and \( 2.54 \) from both sides:

$$ 0.3x^2 - 4.26x + 13.923 = 0 $$

Step 5: Multiply through by 1000 to eliminate decimals (optional, but easier)

$$ 300x^2 - 4260x + 13923 = 0 $$

We can simplify by dividing by 3:

$$ 100x^2 - 1420x + 4641 = 0 $$

Step 6: Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a = 100 \), \( b = -1420 \), \( c = 4641 \)

First, calculate the discriminant \( \Delta \):

$$ \Delta = b^2 - 4ac = (-1420)^2 - 4(100)(4641) $$

$$ \Delta = 2016400 - 1856400 = 160000 $$

Then, find \( x \):

$$ x = \frac{1420 \pm \sqrt{160000}}{200} = \frac{1420 \pm 400}{200} $$

Step 7: Calculate the two solutions for \( x \)

First solution:

$$ x = \frac{1420 + 400}{200} = \frac{1820}{200} = 9.1 $$

Second solution:

$$ x = \frac{1420 - 400}{200} = \frac{1020}{200} = 5.1 $$

Step 8: Find the corresponding \( y \)-values

For \( x = 9.1 \):

$$ y = 0.6(9.1) + 2.54 = 5.46 + 2.54 = 8 $$

For \( x = 5.1 \):

$$ y = 0.6(5.1) + 2.54 = 3.06 + 2.54 = 5.6 $$

So the intersection points are \( (5.1, 5.6) \) and \( (9.1, 8) \).

Answer:

Les points d'intersection sont \( \boldsymbol{(5.1, 5.6)} \) et \( \boldsymbol{(9.1, 8)} \)