Solve for $$ k $$ $$ 3 k^{2}-10 k+8=0 $$

Question

UpStudy AI · Accepted Answer

We begin with the equation

$$
3 k^{2}-10 k+8=0
$$

**Step 1: Identify the coefficients**

For a quadratic equation in the form

$$
a k^2 + b k + c = 0,
$$

we have:

$$
a = 3,\quad b = -10,\quad c = 8.
$$

**Step 2: Compute the discriminant**

The discriminant $$ \Delta $$ is given by

$$
\Delta = b^2 - 4ac.
$$

Substitute the values of $$ a $$, $$ b $$, and $$ c $$:

$$
\Delta = (-10)^2 - 4(3)(8) = 100 - 96 = 4.
$$

**Step 3: Use the quadratic formula**

The quadratic formula is

$$
k = \frac{-b \pm \sqrt{\Delta}}{2a}.
$$

Substitute $$ b = -10 $$, $$ \Delta = 4 $$, and $$ a = 3 $$:

$$
k = \frac{-(-10) \pm \sqrt{4}}{2(3)} = \frac{10 \pm 2}{6}.
$$

**Step 4: Solve for $$ k $$ for each case**

1. For the plus sign:

$$
k = \frac{10 + 2}{6} = \frac{12}{6} = 2.
$$

2. For the minus sign:

$$
k = \frac{10 - 2}{6} = \frac{8}{6} = \frac{4}{3}.
$$

**Final Answer:**

$$
k = 2 \quad \text{or} \quad k = \frac{4}{3}.
$$
$$ k = 2 $$ or $$ k = \frac{4}{3} $$.

Solve for \( k \) \[ 3 k^{2}-10 k+8=0 \]