The quadratic formula

Statement

Solve the equation $2x^2 - 7x + 3 = 0$ using the quadratic formula.

Solution

With $a=2$, $b=-7$, $c=3$ the discriminant is $\Delta = b^2 - 4ac = (-7)^2 - 4\cdot 2\cdot 3 = 49 - 24 = 25.$ Hence $x = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4},$ so $x_1 = \dfrac{12}{4} = 3$ and $x_2 = \dfrac{2}{4} = \dfrac12$. Check: $2\cdot9 - 7\cdot3 + 3 = 18-21+3 = 0$ and $2\cdot\tfrac14 - 7\cdot\tfrac12 + 3 = \tfrac12 - \tfrac72 + 3 = 0$. Correct. $\boxed{x = 3 \ \vee\ x = \tfrac12}$