Question

deux fonctions dont les règles sont :
y = -0,5(x + 5,9)² + 2,3
y = 2x + 11,6

Explanation:

To solve the system of equations \( y = -0.5(x + 5.9)^2 + 2.3 \) and \( y = 2x + 11.6 \), we set the two expressions for \( y \) equal to each other:

Step 1: Set the equations equal

Since both expressions equal \( y \), we have:

$$ -0.5(x + 5.9)^2 + 2.3 = 2x + 11.6 $$

Step 2: Expand the squared term

First, expand \( (x + 5.9)^2 \):

$$ (x + 5.9)^2 = x^2 + 11.8x + 34.81 $$

Substitute this back into the equation:

$$ -0.5(x^2 + 11.8x + 34.81) + 2.3 = 2x + 11.6 $$

Step 3: Distribute the -0.5

$$ -0.5x^2 - 5.9x - 17.405 + 2.3 = 2x + 11.6 $$

Step 4: Simplify the left side

Combine like terms:

$$ -0.5x^2 - 5.9x - 15.105 = 2x + 11.6 $$

Step 5: Move all terms to the left side

Subtract \( 2x \) and \( 11.6 \) from both sides:

$$ -0.5x^2 - 5.9x - 15.105 - 2x - 11.6 = 0 $$

$$ -0.5x^2 - 7.9x - 26.705 = 0 $$

Step 6: Multiply both sides by -2 to eliminate the decimal coefficient of \( x^2 \)

$$ x^2 + 15.8x + 53.41 = 0 $$

Step 7: Use the quadratic formula

For a quadratic equation \( ax^2 + bx + c = 0 \), the solutions are given by:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Here, \( a = 1 \), \( b = 15.8 \), and \( c = 53.41 \).

First, calculate the discriminant \( D = b^2 - 4ac \):

$$ D = (15.8)^2 - 4(1)(53.41) $$

$$ D = 249.64 - 213.64 $$

$$ D = 36 $$

Now, find the solutions:

$$ x = \frac{-15.8 \pm \sqrt{36}}{2(1)} $$

$$ x = \frac{-15.8 \pm 6}{2} $$

Step 8: Calculate the two solutions

For the plus sign:

$$ x = \frac{-15.8 + 6}{2} = \frac{-9.8}{2} = -4.9 $$

For the minus sign:

$$ x = \frac{-15.8 - 6}{2} = \frac{-21.8}{2} = -10.9 $$

Step 9: Find the corresponding \( y \)-values

For \( x = -4.9 \):

$$ y = 2(-4.9) + 11.6 = -9.8 + 11.6 = 1.8 $$

For \( x = -10.9 \):

$$ y = 2(-10.9) + 11.6 = -21.8 + 11.6 = -10.2 $$

So the solutions to the system are \( (-4.9, 1.8) \) and \( (-10.9, -10.2) \).

Answer:

The solutions are \( \boldsymbol{(-4.9, 1.8)} \) and \( \boldsymbol{(-10.9, -10.2)} \).