The quadratic formula is one of students’ most favourite mathematical formulas. It can be quite easily memorized. Students eagerly use it to solve quadratic equations of the form ax^{2} + bx + c = 0. However, not everyone knows why we use it and where the quadratic formula comes from.

When we are trying to determine the x-intercepts of a parabola represented by a certain quadratic equation, we are solving that equation. Quadratic equations may have 1 solutions, 2 solutions or no solutions. This means that the corresponding parabola has 1, 2 or no x-intercepts. Sometimes, the x-intercepts are not “nice” numbers and it is difficult to use factoring methods to solve an equation to determine the x-intercepts. This is when the quadratic formula could be used.

### Deriving the quadratic formula

Now, let’s see where the quadratic formula comes from, by first solving a general quadratic equation of the form ax^{2} + bx + c = 0, where a is the leading coefficient and c is the constant (the y-intercept of the parabola).

To start, move the constant to the other side of the equation

ax^{2} + bx = -c

Next, divide both parts of the equation by the leading coefficient

(ax)^{2}/a + (bx)/a = (-c)/a

Then, complete the square

x^{2} + 2bx/2a + (b/2a)^{2} = (-c)/a + (b/2a)^{2}

(x + (b/2a))^{2} = b^{2}/4a^{2} – c/a

(x + (b/2a))^{2} = (b^{2} – 4ac)/4a^{2}

The number of roots of this equation depends on the sign of the rational expression (b^{2} – 4ac)/4a^{2}. The denominator of this expression is always positive. Therefore, the sign of the whole expression is determined by its numerator, i.e. b^{2} – 4ac.

The expression b^{2} – 4ac is called the Discriminant of any given quadratic equation of the form ax^{2} + bx + c = 0. It is labeled with the capital letter D.

Therefore, **D = b ^{2} – 4ac** and it is a part of the quadratic formula.

When D is a positive number (greater than 0), then the quadratic equation has two possible solutions and we can continue deriving the quadratic formula as follows:

When D = 0, the equation has only one possible solution (two repeating x-intercepts that are also the vertex of a parabola).

Whereas, when D turns out to be a negative value, the equation has no solutions (the parabola does not have x-intercepts).

### Applying the quadratic formula

Solve an equation 9x^{2} + 6x + 1 = 0.

a = 9, b = 6, c = 1

D = b^{2} – 4ac = (6)^{2} – 4(9)(1) = 0

Since D = 0, we know that the equation will have two identical roots.

x_{1} = (-b + 0)/2a and x_{2} = (-b – 0)2a

x_{1} = -6/18 and x_{2} = -6/18

x_{1,2} = -1/3

Therefore, there is only one repeating solution x_{1,2} = -1/3

## There are no comments yet