Question

. Find the zeros of the quadratic equation y=6x2+13x+5
. You may use any of the following methods: factoring, square roots, complete the square, or quadratic formula.

Please show all steps to solve these two questions. Upload a picture of your work for each and clearly label them as #2 and #3. You may use Desmos to check your work, but you must upload your handwritten work to receive credit.

Answer 1

To find the zeros of the quadratic equation \(y = 6x^2 + 13x + 5\), we can use the quadratic formula:

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

where \(a = 6\), \(b = 13\), and \(c = 5\).

### Step 1: Calculate the discriminant

First, we need to calculate the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = 13^2 - 4(6)(5)
\]
\[
= 169 - 120
\]
\[
= 49
\]

### Step 2: Apply the quadratic formula

Now we can substitute \(a\), \(b\), and the discriminant into the quadratic formula:

\[
x = \frac{{-13 \pm \sqrt{49}}}{2(6)}
\]
\[
= \frac{{-13 \pm 7}}{12}
\]

### Step 3: Calculate the two possible values of \(x\)

Now we will find the two solutions:

1. **First solution**:
\[
x_1 = \frac{{-13 + 7}}{12} = \frac{{-6}}{12} = -\frac{1}{2}
\]

2. **Second solution**:
\[
x_2 = \frac{{-13 - 7}}{12} = \frac{{-20}}{12} = -\frac{5}{3}
\]

### Final Result

The zeros of the quadratic equation \(y = 6x^2 + 13x + 5\) are:

\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{3}
\]

You can check this result using Desmos or any graphing tool. Make sure to upload your handwritten work with the clear labels specified.

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I can't upload images or generate handwritten work myself, but you can follow the steps above and write them out on paper, then take a picture to submit.