Lesson 1: Extracting Roots

New Concept

We will learn to solve quadratic equations of the form $ax^2 + c = 0$ by 'extracting roots.' This method involves isolating the $x^2$ term and then taking the square root of both sides to find the solutions.

What’s next

This is just the beginning. Next up, you'll engage with interactive examples and then apply your new skills on a set of practice cards.

Property

A quadratic equation can be written in the standard form

where $a$, $b$, and $c$ are constants, and $a$ is not zero. The numbers $a$, $b$, and $c$ are called parameters and are the coefficients of the quadratic term, the linear term, and the constant term, respectively.

Property

A quadratic equation in two variables has the form

Its graph is not a straight line, but a curve called a parabola. The simplest, or basic, parabola is the graph of $y = x^2$.

Property

To solve a quadratic equation of the form

1. Isolate $x^2$ on one side of the equation.
2. Take the square root of each side.

Property

Extraction of roots can be used to solve some formulas with a quadratic term. To solve for a variable that is squared, first isolate the squared variable term, then take the square root of both sides. Because physical quantities like radius or time must be positive, we often use only the positive square root.

Examples

• The formula for the area of a circle is $A = \pi r^2$. To solve for the radius $r$, first isolate $r^2$ by dividing by $\pi$: $\frac{A}{\pi} = r^2$. Then, take the square root: $r = \sqrt{\frac{A}{\pi}}$.
• The Pythagorean theorem is $a^2 + b^2 = c^2$. To solve for side $a$, first isolate $a^2$: $a^2 = c^2 - b^2$. Then, take the square root: $a = \sqrt{c^2 - b^2}$.
• The formula for the volume of a cone is $V = \frac{1}{3}\pi r^2 h$. To solve for $r$, isolate $r^2$ to get $r^2=\frac{3V}{\pi h}$. Then take the square root: $r = \sqrt{\frac{3V}{\pi h}}$.

Explanation

This technique rearranges formulas to find a variable that is squared. It's the same as extracting roots from an equation, but you're working with variables that represent physical quantities, like radius or velocity.