Question

Find the solutions to 2x^2 + 5x - 3 = 0

Answer 1

To solve the quadratic equation \(2x^2 + 5x - 3 = 0\), we can use the quadratic formula, which is given by:

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

In this equation, \(a = 2\), \(b = 5\), and \(c = -3\). Now, we will calculate the discriminant (\(b^2 - 4ac\)) first:

\[
b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49
\]

Now that we have the discriminant, we can substitute it into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{49}}{2 \cdot 2}
\]

Calculating the square root of 49:

\[
\sqrt{49} = 7
\]

Now we can substitute this value back into the formula:

\[
x = \frac{-5 \pm 7}{4}
\]

This gives us two possible solutions:

1. For the positive case:

\[
x = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}
\]

2. For the negative case:

\[
x = \frac{-5 - 7}{4} = \frac{-12}{4} = -3
\]

Thus, the solutions to the equation \(2x^2 + 5x - 3 = 0\) are:

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