Discriminant of a quadratic polynomial

Instructions:

Write the coefficients of the quadratic equation in boxes \(a\), \(b\), and \(c\) to change the polynomial.

Explanation

By completing a square, it is possible to deduce one of the most important formulas in algebra, the quadratic formula for the standard cuadratic equation \(a > 0\). To do this, it will be necessary to solve for \(x\).

$$ax^2 + bx + c = 0$$

We subtract the constant term \(c\) from both sides of the equation.

$$\begin{split} ax^2 + bx + c - c &= 0 - c \\ ax^2 + bx &= -c \end{split}$$

We divide both sides of the equation by the leading coefficient \(a\).

$$\begin{split} \frac{ax^{2} + bx}{a} &= -\frac{c}{a} \\ x^2 + \frac{b}{a}x &= -\frac{c}{a} \end{split}$$

We complete the square by dividing the coefficient of \(x\) by \(2\) and squaring the result.

$$\begin{split} \frac{\frac{b}{a}}{2} &= \frac{b}{2a} \\ \left(\frac{b}{2a}\right)^{2} &= \frac{b^{2}}{4a^{2}} \end{split}$$

We add this amount to both sides of the equation

$$x^{2} + \frac{b}{a}x + \frac{b^{2}}{4a^{2}} = \frac{b^{2}}{4a^{2}} - \frac{c}{a}$$

Since \(\frac{c}{a}\) is equivalent to \(\frac{4ac}{4a^{2}}\), we simplify.

$$\frac{b^{2}}{4a^{2}} - \frac{c}{a} = \frac{b^{2} - 4ac}{4a^{2}}$$

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

Note that the left-hand side is equivalent to the perfect square trinomial

$$\left( x + \frac{b}{2a} \right)^2 = x^{2} + 2(x)\left( \frac{b}{2a} \right) + \left( \frac{b}{2a} \right)^{2} = x^{2} + \frac{b}{a}x + \frac{b^{2}}{4a^{2}}$$

The left side is factored

$$\left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}$$

Use square root property

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

Quotient rule for square roots

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

\(\frac{b}{2a}\) is substracted

$$\begin{split} x + \frac{b}{2a} - \frac{b}{2a} &= - \frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}} \\ x &= -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \end{split}$$

The like terms are reduced

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