Solve by completing the square is a method for solving linear equations. Linear equations are equations that involve values instead of variables.
Completing the square is used to solve linear equations where one side of the equation is a constant value, or 0. When solving linear equations where neither side of the equation is 0, other methods can be used.
When completing the square, you add an extra row underneath the original grid of numbers. You then combine these numbers to find a new number, which is added to either side of the original equation.
This new number is used to solve the original equation by either being substituted for either variable or being added or subtracted from both sides of the equation. After this, you check to see if your solution is true or false.
Solve the equation by completing the square
Now that you know how to solve for the solution set of the equation, let’s solve this equation by completing the square.
First, we need to find all of the solutions of the equation. To do this, we need to find where the line x = 12 and -15 intersect. Once we find that point, we can add it to both sides of the equation and get x = 0. This will tell us that there is no x-value that satisfies the equation.
Now we can add 0 to both sides of the equation and get y = 12. This tells us that y = 12 is a solution of the equation. We can also add any number less than 0 to both sides of the equation and get y numbers less than 0 are solutions of the equation.
Find zeros using complex numbers
A complex number is a number that has a real part and an imaginary part. These are usually written as x + yi, where i is the imaginary unit, which is defined as x2 = -1.
Complex numbers are usually used to solve equations that have linear variables, like x2 = y2. In these cases, you can replace the variable with its square and then solve it.
For example, if you had the equation x + 2 = 0, you could write this as (x + 2)+(0-2)i=0. Then you could solve this by finding what value of i makes it true.
You could also use complex numbers to find the solution set of an equation using completing the square. First, divide both sides of the equation by a negative number to get rid of the negative sign.
Solve the equation by factoring
Another way to solve the equation is to factor it. Factorization is the process of finding terms that when multiplied together would equal the original term.
In this case, you would find two terms that when multiplied together would equal 12x – 15. These two terms would be x(12 – x) and (x + 5)(x – 1).
Then, you would multiply each of these by a positive or negative one, depending on the original variable. This would give you two new solutions: one with a positive x and one with a negative x.
These are both valid solutions, so there are actually two solutions to the equation! One solution set contains one solution with a positive x and one with a negative x, and the other solution set contains 12 – 15 as the only solution in that set.
Find zeros using Newton’s method
Now that we have our solution set, let’s find the zeros of the function. Zeros are the solutions to x=0, where x is the variable in our equation.
To find the zeros, we’ll use Newton’s method, which is an algorithm for finding roots, or zeros, of a function. Newton’s method can be used for both linear and non-linear functions.
How does it work? Well, it starts with a guess that is fairly close to the zero you are looking for. Then, it calculates a new guess based on the previous one by making slight adjustments — called derivatives.
We will use our solution set from above: {−4|−2|2}. We will start with −4 as our guess and then calculate our derivatives: −4′=−4+2=2Given that 2 is in our solution set but not in the original problem, this new guess is not correct — but it will get us closer.
We keep doing this until we get a number that is not in our solution set.
In this case, after several iterations we end up with −1 as the closest number to 0 in the solutions set.
So what did we learn? The solution set contains all possible answers except for one!
More often than not your answer will contain an infinite number of answers — there are no guarantees your answer will be finite.
This article was written by Stanford mathematics professor Keith Foulk; reprinted with his permission.
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Solve the equation by quadratic equations
Quadratic equations are equations with the second-degree polynomial as the variable. The most common quadratic equation is x2 = y, where x is equal to y squared.
Quadratic equations can be solved in two ways: by completing the square and by factorization. Completing the square is a more advanced method, but it can be used to solve any quadratic equation, not just ones that can be factorized.
To solve x2 = y by completing the square, first write the equation in the form Ax=B, where A is the coefficient of x and B is the constant that x is equal to. Then, find a number a such that a2=B. Then, add (a/2)² to both sides of the equation.
Know where to find solutions of linear equations
Now that you know how to solve linear equations, you should also know where to find the solutions. The solution set of a linear equation is all of the values that solve the equation.
Linear equations can have one solution, infinite solutions, or no solutions. One solution means that one value or set of values solves the equation. Infinite solutions means there are an infinite number of values that solve the equation. No solutions means that there are no values that solve the equation.
Solutions can be real numbers, imaginary numbers, or finite sets of numbers such as a sequence. Real numbers are just things like 2, -2, 4, 0, etc. Imaginary numbers are things like i, -1i, 2i, etc. (Just make up a number then add a i to the end of it.
