Solve 2x² − 5x − 3 = 0

Step-by-step solution

1. 1

   Given equation

   We are given the quadratic equation:

   $$2x^2 - 5x - 3 = 0$$
2. 2

   Quadratic formula

   For $ax^2 + bx + c = 0$, the roots are given by the quadratic formula:

   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
3. 3

   Identify coefficients

   Compare the given equation with the standard form to identify $a, b, c$:

   $$a=2, b=-5, c=-3$$
4. 4

   Substitute values

   Substitute these values into the quadratic formula:

   $$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}$$
5. 5

   Simplify discriminant

   Simplify the expression under the square root (the discriminant):

   $$x = \frac{5 \pm \sqrt{25 - (-24)}}{4}$$
6. 6

   Further simplification

   Continue simplifying the expression:

   $$x = \frac{5 \pm \sqrt{49}}{4}$$
7. 7

   Calculate square root

   Calculate the square root of 49:

   $$x = \frac{5 \pm 7}{4}$$
8. 8

   Find two roots

   Separate into two cases to find the two roots:

   $$x_1 = \frac{5 + 7}{4}, x_2 = \frac{5 - 7}{4}$$
9. 9

   Final roots

   Calculate the values for $x_1$ and $x_2$:

   $$x_1 = 3, x_2 = -1/2$$