The function below has at least one rational zero.
Use this fact to find all zeros of the function.
If there is more than one zero, separate them with commas. Write exact values, not decimal approxima
In

Question

UpStudy AI · Accepted Answer

We are given the quadratic function

$$
12x^2 + 12x + 7.
$$

We use the quadratic formula. For a quadratic equation

$$
ax^2 + bx + c = 0,
$$

the zeros are

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

In our case, $$a=12$$, $$b=12$$, and $$c=7$$.

1. First, compute the discriminant:

$$
b^2 - 4ac = 12^2 - 4(12)(7) = 144 - 336 = -192.
$$

2. The discriminant is negative, so the zeros will be complex. Next, simplify the square root:

$$
\sqrt{-192} = \sqrt{192}\,i.
$$

Note that

$$
192 = 64 \times 3 \quad \Longrightarrow \quad \sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}.
$$

Thus,

$$
\sqrt{-192} = 8\sqrt{3}\,i.
$$

3. Substitute into the quadratic formula:

$$
x = \frac{-12 \pm 8\sqrt{3}\,i}{2 \cdot 12} = \frac{-12 \pm 8\sqrt{3}\,i}{24}.
$$

4. Simplify the expression by dividing numerator and denominator by 4:

$$
x = \frac{-3 \pm 2\sqrt{3}\,i}{6}.
$$

We can rewrite this as

$$
x = -\frac{1}{2} \pm \frac{\sqrt{3}}{3}\,i,
$$

since

$$
-\frac{3}{6} = -\frac{1}{2} \quad \text{and} \quad \frac{2}{6} = \frac{1}{3}.
$$

The zeros of the function are

$$
x = -\frac{1}{2} + \frac{\sqrt{3}}{3}\,i,\quad -\frac{1}{2} - \frac{\sqrt{3}}{3}\,i.
$$
The zeros of the function are $$ x = -\frac{1}{2} + \frac{\sqrt{3}}{3}\,i $$ and $$ x = -\frac{1}{2} - \frac{\sqrt{3}}{3}\,i $$.

The function below has at least one rational zero.
Use this fact to find all zeros of the function.
If there is more than one zero, separate them with commas. Write exact values, not decimal approxima
In

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The function below has at least one rational zero. Use this fact to find all zeros of the function. If there is more than one zero, separate them with commas. Write exact values, not decimal approxima In

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Solution

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The function below has at least one rational zero.
Use this fact to find all zeros of the function.
If there is more than one zero, separate them with commas. Write exact values, not decimal approxima
In