Question

find the zeros of the function $f(x) = 1.6x^2 - 8.3x + 4$. round values to the nearest thousandth (if necessary).
answer attempt 1 out of 2
add a zero no zeros

Explanation:

To find the zeros of the quadratic function \( f(x) = 1.6x^2 - 8.3x + 4 \), we use the quadratic formula. For a quadratic equation \( ax^2 + bx + c = 0 \), the solutions (zeros) are given by:

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

Step 1: Identify \( a \), \( b \), and \( c \)

For the function \( f(x) = 1.6x^2 - 8.3x + 4 \), we have:

• \( a = 1.6 \)
• \( b = -8.3 \)
• \( c = 4 \)

Step 2: Calculate the discriminant \( D = b^2 - 4ac \)

$$ D = (-8.3)^2 - 4(1.6)(4) $$

$$ D = 68.89 - 25.6 $$

$$ D = 43.29 $$

Since the discriminant \( D = 43.29 > 0 \), there are two real zeros.

Step 3: Apply the quadratic formula

$$ x = \frac{-(-8.3) \pm \sqrt{43.29}}{2(1.6)} $$

$$ x = \frac{8.3 \pm 6.58}{3.2} $$

Step 4: Calculate the two zeros

For the plus sign:

$$ x_1 = \frac{8.3 + 6.58}{3.2} = \frac{14.88}{3.2} = 4.65 $$

For the minus sign:

$$ x_2 = \frac{8.3 - 6.58}{3.2} = \frac{1.72}{3.2} = 0.5375 \approx 0.538 $$

Answer:

The zeros of the function \( f(x) = 1.6x^2 - 8.3x + 4 \) are \( x \approx 4.65 \) and \( x \approx 0.538 \). So we need to add these zeros. The zeros are \( \boxed{4.65} \) and \( \boxed{0.538} \) (rounded to the nearest thousandth).