Blue Day: Solving Quadratics by Taking the Square Root

A quadratic equation is a polynomial equation of degree 2, typically in the form $$ax^2 + bx + c = 0$$ , where $$a$$ , $$b$$ , and $$c$$ are constants and @@a eq 0@@.

Solving a quadratic equation by taking the square root involves isolating the squared term and then applying the square root to both sides of the equation, which can yield two possible solutions.

The square root property states that if $$x^2 = k$$ , then $$x = \text{±}\sqrt{k}$$ , where $$k$$ is a non-negative number.

To isolate a squared term, you can move all other terms to the opposite side of the equation. For example, in $$(x-1)^2 + 6 = 8$$ , subtract 6 from both sides to get $$(x-1)^2 = 2$$ .

The first step is to take the square root of both sides, leading to $$2x + 1 = \text{±}9$$ .

The solutions are $$x = \text{±}4$$ .

First, add 4 to both sides to get $$5b^2 = 45$$ , then divide by 5 to find $$b^2 = 9$$ , and finally take the square root to get $$b = \text{±}3$$ .

The discriminant, given by $$b^2 - 4ac$$ , determines the nature of the roots: if positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.

It means that the graph of the quadratic does not intersect the x-axis, indicating that the roots are complex or imaginary.

The solution can be expressed as $$x = \text{±}5$$ .