How to do complete the square step by step?

In mathematics, completing the square is a technique used to solve quadratic equations. It involves transforming a quadratic equation into a perfect square trinomial, which can then be easily solved. Completing the square is especially useful when dealing with equations that are not easily factorizable.

To complete the square in a quadratic equation, follow these step-by-step instructions:

1. Step 1: Rewrite the equation in the form ax^2 + bx + c = 0. Ensure that the coefficient of the x^2 term is 1. If necessary, divide through all terms by this coefficient.
2. Step 2: Move the constant term (c) to the right side of the equation, making sure the left side only contains terms with x.
3. Step 3: Take half of the coefficient of the x term (b/2), square it, and add this value to both sides of the equation. This step aims to create a perfect square trinomial on the left side.
4. Step 4: Factor the left side of the equation as a perfect square trinomial. It should be in the form of (x + p)^2, where p is the value obtained in the previous step.
5. Step 5: Rewrite the equation using the factored form obtained in the previous step.
6. Step 6: Solve the equation by taking the square root of both sides and isolating x.
7. Step 7: Check the solution by substituting it back into the original equation.

By following these steps, you can complete the square to solve quadratic equations efficiently. Understanding this technique is essential in algebra and can help in various mathematical applications.

How do you complete the square steps?

In mathematics, completing the square is a technique used to manipulate and solve quadratic equations. It involves transforming a quadratic equation into a perfect square trinomial. The process can be broken down into several steps.

The first step is to make sure the coefficient of the quadratic term is 1. If it isn't, divide the entire equation by the coefficient to simplify it. This step ensures that the equation is in the standard form (x - h)^2 = k, where h and k are constants.

Next, take half of the coefficient of the linear term, square it, and add it to both sides of the equation. This completes the square on the left side of the equation. The purpose of this step is to create a perfect square trinomial, which can be factored easily.

Now that the left side of the equation is a perfect square, it can be factored as (x - h)^2. To find the value of h, take the square root of the constant term on the right side of the equation. The resulting expression is x - h = sqrt(k).

The final step is to solve for x by adding h to both sides of the equation. This gives you the two possible solutions for x:

x = h +- sqrt(k)

Completing the square is particularly useful when solving quadratic equations that cannot be factored easily or when finding the vertex of a parabola. It helps simplify complex equations and provides a straightforward method for finding solutions.

What is the formula for a perfect square?

What is the formula for a perfect square? A perfect square is a number that can be expressed as the square of an integer. This means that a perfect square is the product of an integer multiplied by itself.

So, what is the formula? The formula for a perfect square is simply to multiply an integer by itself. Mathematically, it can be represented as:

x^2 = x * x

Here, x represents the integer.

For example, if we want to find the perfect square of 4, we can use the formula and multiply 4 by itself. The result is 16, which means that 16 is a perfect square.

Similarly, if we want to find the perfect square of 9, we can use the formula again and multiply 9 by itself. The result is 81, so 81 is a perfect square.

It is important to note that a perfect square can only be a positive number. This is because when we multiply a positive number by another positive number, the result will always be positive.

Another key point to remember is that every perfect square has two square roots. The square root of a number is the value that, when multiplied by itself, gives the original number. For example, the square roots of 16 are 4 and -4, because both 4 and -4, when squared, equal 16.

Using the formula for perfect squares can be helpful in various mathematical applications, including solving equations, finding the sides of a square, and determining the area of a square. By understanding the concept of perfect squares and their formula, we can enhance our mathematical skills and solve various problems with ease.

How do you factor by completing the square?

In algebra, when we have a quadratic equation in the form of ax^2 + bx + c, we can use the method of completing the square to factorize it.

To start factoring by completing the square, we first need to ensure that the coefficient of x^2 is 1. If it is not, we can factor out the coefficient from all terms to make it 1.

Next, we move the constant term c to the other side of the equation and create a square by halving the coefficient of x and squaring it. This means adding (b/2)^2 to both sides of the equation.

By doing this, we obtain a perfect square trinomial on the left side of the equation. We can then rewrite the trinomial as the square of a binomial. The binomial will be the factors we are looking for.

Finally, we take the square root of both sides of the equation and solve for x. This will give us the two possible values for x that make the equation true.

By utilizing the completing the square method, we are able to factorize quadratic equations and express them in the form of (x + a)(x + b), where a and b are the solutions obtained from the square root step.

This method is particularly useful when dealing with complex quadratic equations that cannot be easily factored using other techniques, such as factoring by grouping or using the quadratic formula.

How to solve equations step by step?

To solve equations step by step, it is important to follow a systematic approach. Here is a detailed guide on how to solve equations:

Step 1: Start by identifying the given equation. Determine the unknown variable and ensure that the equation is in a suitable form for solving.

Step 2: Simplify the equation by combining like terms and applying the necessary operations. This often involves performing addition, subtraction, multiplication, or division on both sides of the equation.

Step 3: Use the appropriate properties of equality to manipulate the equation further. These properties include the reflexive, symmetric, transitive, addition, subtraction, multiplication, and division properties.

For example, if you have the equation 2x + 5 = 17, you can subtract 5 from both sides to get 2x = 12.

Step 4: Isolate the variable by performing inverse operations. If the variable is being added, subtract it from both sides. If the variable is being multiplied, divide both sides by the coefficient.

Continuing with the previous example, you can divide both sides of the equation 2x = 12 by 2, resulting in x = 6.

Step 5: Check your solution by substituting the value of the variable back into the original equation. Ensure that both sides of the equation are equal after the substitution.

This step is important to verify if your solution is correct. In the example above, substitute x = 6 back into the equation 2x + 5 = 17 and confirm that both sides equal to 17.

Step 6: If the equation has more variables or parameters, repeat the process for each unknown until all the variables are solved.

Remember, practice is key when it comes to solving equations. The more you practice, the more comfortable you will become with the steps involved in solving different types of equations.