Question

If \(\int_{0}^{3} (3x^2-4x+2) \,dx = k,\) then an integer root of 3x²-4x+2= \(\frac{3k}{5}\) is

• A. 1
• B. 0
• C. 15
• D. -1

Answer 1

The Correct Option is D

To solve the problem, we need to evaluate the integral $I = \int_0^3 (3x^2 - 4x + 2) dx$, set it equal to $k$, and find an integer root of the equation $3x^2 - 4x + 2 = \frac{3k}{5}$.

1. Evaluate the Integral:
Compute $I = \int_0^3 (3x^2 - 4x + 2) dx$.
Antiderivative: $\int (3x^2 - 4x + 2) dx = x^3 - 2x^2 + 2x$.
Evaluate at the limits:
$\left[ x^3 - 2x^2 + 2x \right]_0^3 = (3^3 - 2(3^2) + 2(3)) - 0 = 27 - 18 + 6 = 15$.
Thus, $k = 15$.

2. Form the Equation:
Substitute $k = 15$ into $3x^2 - 4x + 2 = \frac{3k}{5}$:
$\frac{3k}{5} = \frac{3 \cdot 15}{5} = 9$.
So, $3x^2 - 4x + 2 = 9$, which simplifies to:
$3x^2 - 4x + 2 - 9 = 0$, or $3x^2 - 4x - 7 = 0$.

3. Solve for the Integer Root:
Solve $3x^2 - 4x - 7 = 0$ using factoring:
Find numbers whose product is $3 \cdot (-7) = -21$ and sum is $-4$. These are $-7$ and $3$.
Rewrite:
$3x^2 - 7x + 3x - 7 = 0$.
Group:
$x(3x - 7) + (3x - 7) = 0$, so $(3x - 7)(x + 1) = 0$.
Roots: $3x - 7 = 0 \implies x = \frac{7}{3}$, or $x + 1 = 0 \implies x = -1$.
The integer root is $x = -1$.

Final Answer:
The integer root is $-1$.

Answer 2

Step 1: Evaluate the Integral

We are given the integral to evaluate:

I = ∫₀³ (3x² - 4x + 2) dx

First, let's find the antiderivative of the integrand 3x² - 4x + 2. The antiderivative is:

∫ (3x² - 4x + 2) dx = x³ - 2x² + 2x

Now, evaluate the integral at the limits from 0 to 3:

[ x³ - 2x² + 2x ]₀³ = (3³ - 2(3²) + 2(3)) - 0 = 27 - 18 + 6 = 15

Thus, we find that k = 15.

Step 2: Form the Equation

Next, we substitute k = 15 into the equation 3x² - 4x + 2 = (3k / 5).

Substituting for k, we get:

(3k / 5) = (3 * 15) / 5 = 9

Now, substitute this value into the equation:

3x² - 4x + 2 = 9

Simplifying this equation:

3x² - 4x + 2 - 9 = 03x² - 4x - 7 = 0

Now, we need to solve this quadratic equation.

Step 3: Solve for the Integer Root

We need to solve the quadratic equation 3x² - 4x - 7 = 0 by factoring.

First, look for two numbers whose product is 3 * (-7) = -21 and whose sum is -4. These numbers are -7 and 3.

Now, rewrite the middle term of the equation:

3x² - 7x + 3x - 7 = 0

Group the terms:

x(3x - 7) + (3x - 7) = 0

Factor out the common factor (3x - 7):

(3x - 7)(x + 1) = 0

Now, solve for the roots:

3x - 7 = 0 → x = 7 / 3x + 1 = 0 → x = -1

The integer root is x = -1.

Final Answer:

The integer root of the equation 3x² - 4x - 7 = 0 is x = -1.

Conclusion:

We have successfully evaluated the integral and solved for the integer root of the equation. The key steps involved integrating the function, substituting the result, and solving the quadratic equation by factoring. This process is essential for understanding integration and solving algebraic equations in calculus.