To determine the zeros of the quadratic function, we start with the equation:
y equals six x squared plus thirteen x plus five.
The zeros of a quadratic function can be found using the quadratic formula, which states that for any quadratic in the standard form Ax squared plus Bx plus C equals zero, the solutions for x can be found using:
x equals negative B plus or minus the square root of B squared minus four A C, all divided by two A.
In our case, A is six, B is thirteen, and C is five.
First, we need to calculate B squared minus four A C.
We first find B squared, which is thirteen squared. Thirteen squared equals one hundred sixty-nine.
Next, we calculate four times A times C. The value of A is six, and the value of C is five, so:
Four times six times five equals one hundred twenty.
Now we can substitute these values into the expression for the discriminant:
B squared minus four A C equals one hundred sixty-nine minus one hundred twenty.
When we perform the subtraction, we get forty-nine.
Now we substitute these values into the quadratic formula. First, we need to calculate negative B. Since B is thirteen, negative B equals negative thirteen.
Now we have:
x equals negative thirteen plus or minus the square root of forty-nine, all divided by two A.
We know that the square root of forty-nine is seven. Therefore, we can update our equation to:
x equals negative thirteen plus or minus seven, divided by two times six.
Now we calculate two times six, which equals twelve, so our equation becomes:
x equals negative thirteen plus or minus seven, divided by twelve.
Now we will find the two possible values for x by considering both the plus and minus.
First, we take the plus case:
x equals negative thirteen plus seven, divided by twelve.
Negative thirteen plus seven equals negative six, so we have:
x equals negative six divided by twelve.
When we simplify negative six divided by twelve, we find:
x equals negative one half.
Now, we consider the minus case:
x equals negative thirteen minus seven, divided by twelve.
Negative thirteen minus seven equals negative twenty, so we have:
x equals negative twenty divided by twelve.
When we reduce negative twenty divided by twelve, we find:
x equals negative five thirds.
Thus, the zeros of the quadratic function y equals six x squared plus thirteen x plus five are:
x equals negative one half and x equals negative five thirds.