Q. \(2x^2 – x – 3 = 0\)

We want to solve the equation

\[
2x^2 – x – 3 = 0
\]

Step 1: Identify the coefficients.

This is a quadratic equation in the standard form

\[
ax^2 + bx + c = 0
\]

Here:

\[
a = 2,\quad b = -1,\quad c = -3
\]

Step 2: Use the quadratic formula.

The quadratic formula is

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

Step 3: Substitute \(a\), \(b\), and \(c\).

Substitute \(a = 2\), \(b = -1\), \(c = -3\):

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 – 4(2)(-3)}}{2(2)}
\]

Step 4: Simplify inside the square root.

First compute the discriminant:

\[
b^2 – 4ac = (-1)^2 – 4(2)(-3)
\]

Compute each part:

\[
(-1)^2 = 1
\]

\[
-4(2)(-3) = -8(-3) = 24
\]

So the discriminant becomes:

\[
1 + 24 = 25
\]

Step 5: Continue simplifying the quadratic formula.

Now we have:

\[
x = \frac{1 \pm \sqrt{25}}{4}
\]

Since

\[
\sqrt{25} = 5
\]

we get:

\[
x = \frac{1 \pm 5}{4}
\]

Step 6: Split into two solutions.

First solution using the plus sign:

\[
x = \frac{1 + 5}{4} = \frac{6}{4} = \frac{3}{2}
\]

Second solution using the minus sign:

\[
x = \frac{1 – 5}{4} = \frac{-4}{4} = -1
\]

Final Answer:

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