The Quadratic Formula: A Reliable Tool for Solving Second‑Degree Equations

When faced with a quadratic equation of the form ax² + bx + c = 0, the most systematic approach is to use the quadratic formula. The formula arises naturally from the algebraic technique known as completing the square, and it guarantees that every second‑degree equation has at least one real or complex solution.

By isolating the variable and manipulating the equation, we arrive at the celebrated result:

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

Here, x denotes the roots of the equation, while a, b, and c are the numerical coefficients that define the specific quadratic. The expression under the square root, b² – 4ac, is called the discriminant; it tells us whether the roots are real and distinct (discriminant > 0), real and equal (discriminant = 0), or complex conjugates (discriminant < 0).

Example:

Let us solve the quadratic equation 6x² + 11x – 35 = 0 using the formula.

First, identify the coefficients: a = 6, b = 11, c = –35.

Compute the discriminant: $$\Delta = b^2 - 4ac = 11^2 - 4(6)(-35) = 121 + 840 = 961$$ Since \Delta is a perfect square (\sqrt{961} = 31), the roots will be rational.

Apply the quadratic formula: $$x_{1} = \frac{-11 + 31}{2(6)} = \frac{20}{12} = \frac{5}{3}$$ $$x_{2} = \frac{-11 - 31}{2(6)} = \frac{-42}{12} = -\frac{7}{2}$$ Thus, the equation has two real solutions: x = \tfrac{5}{3} and x = -\tfrac{7}{2}.

This method works for any quadratic, regardless of whether its roots are integers, fractions, or complex numbers. By mastering the quadratic formula, you gain a powerful tool for algebraic problem‑solving.