The x-intercepts of $$ y=3 x^{2}-8 x+1 $$ are 0.13 and 2.54 Select one: True False

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Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$3x^{2}-8x+1=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{8\pm \sqrt{\left(-8\right)^{2}-4\times 3}}{2\times 3}$$
- step2: Simplify the expression:
  $$x=\frac{8\pm \sqrt{\left(-8\right)^{2}-4\times 3}}{6}$$
- step3: Simplify the expression:
  $$x=\frac{8\pm \sqrt{52}}{6}$$
- step4: Simplify the expression:
  $$x=\frac{8\pm 2\sqrt{13}}{6}$$
- step5: Separate into possible cases:
  $$\begin{align}&x=\frac{8+2\sqrt{13}}{6}\&x=\frac{8-2\sqrt{13}}{6}\end{align}$$
- step6: Simplify the expression:
  $$\begin{align}&x=\frac{4+\sqrt{13}}{3}\&x=\frac{8-2\sqrt{13}}{6}\end{align}$$
- step7: Simplify the expression:
  $$\begin{align}&x=\frac{4+\sqrt{13}}{3}\&x=\frac{4-\sqrt{13}}{3}\end{align}$$
- step8: Rewrite:
  $$x_{1}=\frac{4-\sqrt{13}}{3},x_{2}=\frac{4+\sqrt{13}}{3}$$
To determine if the x-intercepts of the equation $$ y = 3x^2 - 8x + 1 $$ are 0.13 and 2.54, we need to solve the equation $$ 3x^2 - 8x + 1 = 0 $$.

The solutions to this quadratic equation are given by:

$$
x_{1} = \frac{4 - \sqrt{13}}{3}, \quad x_{2} = \frac{4 + \sqrt{13}}{3}
$$

Now, let's calculate these values to see if they match the given x-intercepts of 0.13 and 2.54.

1. Calculate $$ x_{1} $$:
   $$
   x_{1} = \frac{4 - \sqrt{13}}{3}
   $$

2. Calculate $$ x_{2} $$:
   $$
   x_{2} = \frac{4 + \sqrt{13}}{3}
   $$

Now, we will compute these values.

First, we find $$ \sqrt{13} $$ which is approximately 3.60555.

Now substituting this value:

1. For $$ x_{1} $$:
   $$
   x_{1} = \frac{4 - 3.60555}{3} \approx \frac{0.39445}{3} \approx 0.13148
   $$

2. For $$ x_{2} $$:
   $$
   x_{2} = \frac{4 + 3.60555}{3} \approx \frac{7.60555}{3} \approx 2.53518
   $$

Now we compare these results with the given x-intercepts:

- $$ x_{1} \approx 0.13148 $$ (which is approximately 0.13)
- $$ x_{2} \approx 2.53518 $$ (which is approximately 2.54)

Since both calculated x-intercepts are very close to the given values, we can conclude that the statement is **True**.
The statement is True.

The x-intercepts of \( y=3 x^{2}-8 x+1 \) are 0.13 and 2.54 Select one: True False

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