- Square Root Property
- We have been using only principal square roots,
but now we'll start using both the positive and negative square root.
- The square root property is that if
, where a
is a real number, then
||Solve (x - 3)2 = 4.
||x - 3 = ±2
||Square root property
||x = 3 ± 2
||Add 3 to both sides
||x = 3 + 2 or
x = 3 - 2
||x = 5, 1
- Idea of Completing the Square
- Given the problem
Solve x2 + bx + c = 0,
find a number that complets the square of x2 + bx to get to an equivalent equation of the form
Solve (x - d)2 = f,
that can be solved as in the above example.
- Steps to Solving a Quadratic Equation by Completing the Square
- Make the numerical coefficient of the x2 term equal to 1.
- Rewrite the equation with the constant by itself on the right side of the equation.
- Take ½ the numerical coefficient of the x term, square it, and add this
quantity to both sides of the equation.
- Factor the trinomial into the square of a binomial.
- Use the square root property to take the square root of both sides of the equation.
- Solve for x.
- Check the solution in the original equation.
- Note that the variable was called "x" in the above steps, but it can be called by any variable name, not necessarily just x.