Question

question
find the zeros of the function ( f(x) = 2x^2 + 10x + 6 ). round values to the nearest thousandth (if necessary).
answer
attempt 1 out of 2
+ add a zero - no zeros
( x = ) submit answer

Explanation:

Step1: Recall Quadratic Formula

For a quadratic function \( f(x) = ax^2 + bx + c \), the zeros are given by \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 2 \), \( b = 10 \), \( c = 6 \).

Step2: Calculate Discriminant

First, find the discriminant \( D=b^2 - 4ac \). Substitute the values: \( D = 10^2-4\times2\times6=100 - 48 = 52 \).

Step3: Apply Quadratic Formula

Now, substitute \( a \), \( b \), and \( D \) into the quadratic formula: \( x=\frac{-10\pm\sqrt{52}}{2\times2}=\frac{-10\pm2\sqrt{13}}{4}=\frac{-5\pm\sqrt{13}}{2} \).

Step4: Compute Numerical Values

Calculate the two roots:

• For the plus sign: \( x=\frac{-5+\sqrt{13}}{2}\approx\frac{-5 + 3.60555}{2}=\frac{-1.39445}{2}\approx - 0.697 \)
• For the minus sign: \( x=\frac{-5-\sqrt{13}}{2}\approx\frac{-5 - 3.60555}{2}=\frac{-8.60555}{2}\approx - 4.303 \)

Answer:

The zeros of the function are \( x\approx - 0.697 \) and \( x\approx - 4.303 \). (If we need to fill one at a time, we can present them separately. For example, if filling the first one, \( x\approx - 0.697 \); if the second, \( x\approx - 4.303 \))