Question

What formula can you use to solve any quadratic equation?

Short answer

To solve any quadratic equation, the quadratic formula, which is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), can be used.

Step-by-step solution

Identify the Quadratic Equation

First, it is important to identify the quadratic equation which has the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) represents variables, and \(x\) is the unknown variable. The coefficients \(a\), \(b\), and \(c\) are the quadratic, linear, and constant terms, respectively.

The Quadratic Formula

The formula to resolve this type of equation is known as the quadratic formula and it's given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates both possible values for \(x\) (if they exist) by appending the plus-minus sign (\(\pm\)) before the square root.

Explanation of the Quadratic Formula

The terms inside the square root, \(b^2 - 4ac\), is known as the discriminant. The discriminant can predict the types of roots the equation has. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (or a repeated real root). If the discriminant is negative, the equation has two complex roots.

Applying the Quadratic Formula

When you have a quadratic equation in the form of \(ax^2 + bx + c = 0\), you can solve it by simply plugging the coefficients \(a\), \(b\), and \(c\) into the quadratic equation and solving it for \(x\).