Question

section 3.5 homework

1. solve the system by elimination.

$y = 9x^2 + 6x + 2$
$y = x^2 - 8x - 19$

a. $square$ $(0, 0)$
b. $square$ $(2, 6)$
c. $square$ $(8, -12)$
d. $square$ $(4, 0)$
e. $square$ $(4, -1)$
f. $square$ no real solution

Explanation:

Step1: Set the equations equal

Since both equations are solved for \( y \), set \( 9x^2 + 6x + 2 = x^2 - 8x - 19 \).

Step2: Simplify the equation

Subtract \( x^2 - 8x - 19 \) from both sides: \( 9x^2 - x^2 + 6x + 8x + 2 + 19 = 0 \), which simplifies to \( 8x^2 + 14x + 21 = 0 \).

Step3: Calculate the discriminant

For a quadratic \( ax^2 + bx + c = 0 \), the discriminant is \( D = b^2 - 4ac \). Here, \( a = 8 \), \( b = 14 \), \( c = 21 \). So \( D = 14^2 - 4(8)(21) = 196 - 672 = -476 \).

Step4: Analyze the discriminant

Since \( D < 0 \), the quadratic has no real solutions, meaning the system has no real solution.

Answer:

f. no real solution