Solve By Using the Quadratic Equation Lessons
The quadratic formula gives us an alternative to Completing the Square when we cannot factor an equation. People often find the Quadratic Formula method easier and more convenient because it does not require many operations on the equation being solved.
For solving an equation in the variable x, the Quadratic Formula is:
To find the solutions to an equation, we simply need to identify what a, b, and c are, then substitute them into this formula, and simplify.
First Example (Two Solutions)
We begin applying the Quadratic Formula by putting the equation in the following form:
This means that each term in the equation must be on the left side, just like when we are factoring or Completing the Square. So we subtract from each side.
Now for consistency, we will rearrange the terms so that they are in the same order: The x2 term first, the x term second, and the constant term last.
Now by comparing our equation with "ax2 + bx + c = 0", we can see that a must equal 1, b must equal 1, and c must equal .
Now that the values of a, b, and c have been determined, we may return to the quadratic formula and use substitution. (Remember to use parentheses when substituting to avoid problems with negative signs.)
We must now simplify this equation keeping the Order of Operations in mind. We begin by simplifying (1)2.
Next, we simplify multiplication. We see that 11 is equal to 15:
Now 1 and 15 are added, resulting in 16.
The square root of 16 is 4.
You may recall that the methods of solving by factoring and solving by completing the square required you to split each problem into multiple subproblems to obtain multiple solutions. Since we are again looking for more than one solution, we must split this problem in two.
So far, we have been carrying the ± sign through the problem. Now, we will create two problems, one with a plus sign, and one with a minus sign.
Simplifying the first subproblem gives
Simplifying the second subproblem gives
We can now combine these two solutions into the solution to the original example problem:
Proceed to the next page for another example.