Question

Square Roots (Level 1) dratic equation for all val 4(x+9)²+8=32

Answer 1

The correct choice is 2) decay.

Step-by-step explanation:

The given expression, 4(x+9)² + 8 = 32, represents a quadratic equation. When a quadratic equation is in the form of \
`(a(x - h)^2 + k\), where \(a\), \(h\), and \(k\)` are constants, the graph of the equation represents a parabola. In this case, the coefficient of the squared term is positive (4), indicating an upward-opening parabola. However, the addition of a constant on the right side of the equation shifts the parabola vertically. The term 8 on the right side causes the parabola to shift downward.

Exponential growth is characterized by an increasing rate, leading to a graph that rises sharply. Quadratic equations, especially those with positive coefficients for the squared term, do exhibit upward curvature, but the addition of a constant term (as in this case) alters this behavior.

On the other hand, exponential decay involves a decreasing rate, leading to a graph that declines sharply. In this quadratic equation, the downward shift of the parabola indicates a decreasing pattern, resembling exponential decay. Therefore, the correct choice is 2) decay, as the function models exponential decay rather than growth, linearity, or pure quadratics.

Answer 2

The solution to the quadratic equation 4(x+9)² + 8 = 32 is x = -11 andx = -7.

Step-by-step explanation:

To find the solution to the given quadratic equation, follow these steps:

1. Expand and Simplify:

Expand and simplify the equation:

`\[4(x+9)^2 + 8 = 32 \implies 4(x^2 + 18x + 81) + 8 = 32\]`

2. Combine Like Terms:

Combine like terms on the right side:

`\[4x^2 + 72x + 324 + 8 = 32 \implies 4x^2 + 72x + 332 = 32\]`

3. Move Constant to One Side:

Move the constant term to one side by subtracting 32 from both sides:

`\[4x^2 + 72x + 332 - 32 = 0 \implies 4x^2 + 72x + 300 = 0\]`

4. Divide by Coefficient:

Divide the entire equation by the coefficient of x², which is 4:

`\[(4x^2 + 72x + 300)/(4) = 0 \implies x^2 + 18x + 75 = 0\]`

5. Factor or Use the Quadratic Formula:

Factor the quadratic expression or use the quadratic formula to find the roots:

`\[(x + 3)(x + 15) = 0 \implies x = -3 \text{ or } x = -15\]`

6.Check Solutions:

However, the original equation involved a square term (x+9)²), and both roots (x = -3 and x = -15 do not satisfy the original equation. Therefore, there are no solutions for these roots.

7. Correct Equation:

Reevaluate the correct equation, considering the square term:(x + 9)²= 0).

`\[(x + 9)^2 = 0 \implies x + 9 = 0 \implies x = -9\]`

8. Final Solution:

The correct solution to the given quadratic equation is \(x = -9\), which satisfies the original equation.