Solving Quadratic Equations by Completing the Square Worksheet, Formula, and Examples

Solve Quadratic Equations by Completing the Square

How to Solve Quadratic Equations by Completing the Square

Completing the square is a useful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. Here are the steps to solve quadratic equations by completing the square:

1. Rewrite the quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.
2. If the coefficient of x^2 is not 1, divide both sides of the equation by a.
3. Move the constant term to the right-hand side of the equation.
4. Divide the coefficient of x by 2 and square the result. Add this value to both sides of the equation.
5. Factor the left-hand side of the equation as a perfect square trinomial.
6. Take the square root of both sides of the equation.
7. Solve for x.
   

Completing the square is particularly useful when the quadratic equation does not factor easily. It is also a useful technique for finding the vertex of a parabola, which is the point where the parabola reaches its maximum or minimum value.

When solving quadratic equations by completing the square, it is important to keep track of the signs. If the coefficient of x^2 is negative, for example, the value inside the square root will be negative, which means there are no real solutions to the equation.

Completing the square can also be used to find the value of a quadratic expression for a given value of x. In this case, the expression is first rewritten in the form ax^2 + bx + c, and then the value of x is substituted into the expression. The resulting expression can then be simplified using the techniques of algebra.

Completing the Square Formula

Completing the square is a method used to solve quadratic equations of the form ax^2 + bx + c = 0. This method involves manipulating the equation so that it can be written in the form (x + p)^2 = q, where p and q are constants. This form is known as a perfect square trinomial.

The goal of completing the square is to find the values of p and q that make the equation true. Once these values are found, the equation can be solved by taking the square root of both sides and solving for x.

To complete the square, the first step is to divide both sides of the equation by the leading coefficient, a. This will give the equation in the form x^2 + (b/a)x + c/a = 0.

Next, the constant term, c/a, is added to both sides of the equation. This will give the equation in the form x^2 + (b/a)x = -c/a.

The next step is to add (b/2a)^2 to both sides of the equation. This will complete the square on the left-hand side of the equation and give the equation in the form (x + b/2a)^2 = (b^2/4a^2) – c/a.

Finally, the equation can be solved by taking the square root of both sides and solving for x. The solution will be in the form x = (-b +/- sqrt(b^2 – 4ac)) / 2a.

The completing the square formula is a useful tool for solving quadratic equations that cannot be easily factored. It is important to note that this method only works for equations with a leading coefficient of 1. If the leading coefficient is not 1, the equation must be divided by the leading coefficient before completing the square.

Solving Quadratic Equations by Completing the Square Examples with Answers

Completing the square is a method used to solve quadratic equations that cannot be factored. The quadratic equation is written in standard form, ax² + bx + c = 0, where a, b, and c are coefficients and x is a variable. The goal is to rewrite the equation in the form of (x – h)² = k, where h and k are constants. Completing the square can be used to find the vertex of a parabola, which is the highest or lowest point on the graph.

Here are some examples of solving quadratic equations by completing the square:

Example 1: Solve the equation 2x² + 8x – 6 = 0 by completing the square.

Step 1: Divide both sides of the equation by the coefficient of x² to get the equation in standard form. 2x² + 8x – 6 = 0 becomes x² + 4x – 3 = 0.

Step 2: Move the constant term to the right side of the equation. x² + 4x = 3.

Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation. (4/2)² = 4, so x² + 4x + 4 = 7.

Step 4: Rewrite the left side of the equation as a perfect square. (x + 2)² = 7.

Step 5: Take the square root of both sides of the equation. x + 2 = ±√7.

Step 6: Solve for x. x = -2 ± √7.

Therefore, the solutions to the equation 2x² + 8x – 6 = 0 are x = -2 + √7 and x = -2 – √7.

Example 2: Solve the equation 3x² – 12x + 7 = 0 by completing the square.

Step 1: Divide both sides of the equation by the coefficient of x² to get the equation in standard form. 3x² – 12x + 7 = 0 becomes x² – 4x + 7/3 = 0.

Step 2: Move the constant term to the right side of the equation. x² – 4x = -7/3.

Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation. (-4/2)² = 4, so x² – 4x + 4 = -7/3 + 4.

Step 4: Rewrite the left side of the equation as a perfect square. (x – 2)² = 5/3.

Step 5: Take the square root of both sides of the equation. x – 2 = ±√(5/3).

Step 6: Solve for x. x = 2 ± √(5/3).

Therefore, the solutions to the equation 3x² – 12x + 7 = 0 are x = 2 + √(5/3) and x = 2 – √(5/3).

Completing the square can be a useful method for solving quadratic equations that cannot be factored. It allows us to rewrite the equation in the form of (x – h)² = k, which can be easily solved by taking the square root of both sides.

Solve by Completing the Square Method

The completing the square method is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. This method involves manipulating the equation by adding or subtracting a constant term to both sides of the equation to create a perfect square trinomial. The goal is to transform the equation into the form (x + p)^2 = q, where p and q are constants.

To solve a quadratic equation using the completing the square method, follow these steps:

1. Move the constant term to the right side of the equation.
2. Divide both sides of the equation by the coefficient of x^2 so that the coefficient of x^2 is 1.
3. Take half of the coefficient of x, square it, and add it to both sides of the equation.
4. Simplify and solve for x.

If the constant term on the right side of the equation is negative, then add the absolute value of the constant term to both sides of the equation instead of subtracting it.

It is important to note that the completing the square method can be used to solve quadratic equations with complex solutions. When the discriminant (b^2 – 4ac) is negative, the solutions are complex numbers. In this case, the square root of a negative number is taken, resulting in an imaginary number. For example, the equation x^2 + 4x + 5 = 0 has no real solutions, but can be solved using the completing the square method to find the complex solutions x = -2 + i and x = -2 – i, where i is the imaginary unit.

Overall, the completing the square method is a powerful technique for solving quadratic equations, even when the solutions are complex. With practice, it can become a valuable tool for any student of algebra.

Completing the Square Steps

Completing the square is a method used to solve quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily solved.

To complete the square, follow these steps:

1. Move the constant term to the right side of the equation.
2. Divide all terms by the coefficient of the squared term, if necessary.
3. Take half of the coefficient of the linear term, square it, and add it to both sides of the equation.
4. Write the left side of the equation as a perfect square trinomial.
5. Solve for the variable.

Let’s look at an example to see these steps in action:

x^2 + 6x + 5 = 0

Step 1: Move the constant term to the right side of the equation.

x^2 + 6x = -5

Step 2: Divide all terms by the coefficient of the squared term, if necessary. In this case, the coefficient is 1, so no division is necessary.

Step 3: Take half of the coefficient of the linear term, square it, and add it to both sides of the equation.

x^2 + 6x + 9 = 4

Step 4: Write the left side of the equation as a perfect square trinomial.

(x + 3)^2 = 4

Step 5: Solve for the variable.

x + 3 = ±2

x = -3 ± 2

x = -1, -5

Completing the square can be especially useful when factoring or using the quadratic formula is not possible or convenient. It can also be used to derive the quadratic formula.

Completing the Square with Coefficient

Completing the square is a useful method for solving quadratic equations. When the quadratic equation has a coefficient, the steps to complete the square are slightly different.

To complete the square of a quadratic equation with a coefficient, one needs to follow these steps:

1. Divide the entire equation by the coefficient to make the coefficient of the squared term equal to 1.
2. Move the constant term to the right-hand side of the equation.
3. Add the square of half the coefficient of the linear term to both sides of the equation.
4. Factor the left-hand side of the equation as a perfect square trinomial.
5. Solve for the variable by taking the square root of both sides of the equation.

For example, consider the quadratic equation 2x^2 + 8x + 3 = 0. To complete the square, divide the entire equation by 2 to get x^2 + 4x + 3/2 = 0. Then, move the constant term to the right-hand side of the equation to get x^2 + 4x = -3/2.

Next, add the square of half the coefficient of the linear term to both sides of the equation. Half of 4 is 2, and the square of 2 is 4. Therefore, add 4 to both sides of the equation to get x^2 + 4x + 4 = 1/2.

Now, factor the left-hand side of the equation as a perfect square trinomial. The left-hand side of the equation can be written as (x + 2)^2. Therefore, the equation becomes (x + 2)^2 = 1/2.

Finally, solve for the variable by taking the square root of both sides of the equation. The solutions are x = -2 + sqrt(1/2) and x = -2 – sqrt(1/2).

Completing the square with coefficient can be used for both binomial and trinomial expressions. The key is to make the coefficient of the squared term equal to 1 before proceeding with the completion of the square.

Comparison with Quadratic Formula

When solving quadratic equations, there are two primary methods: the quadratic formula and completing the square. Both methods can solve any quadratic equation, but they have different strengths and weaknesses.

The quadratic formula is a formula that can be used to solve any quadratic equation in the form of ax^2 + bx + c = 0. The formula is:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

where a, b, and c are coefficients of the quadratic equation. The quadratic formula is a straightforward method of solving quadratic equations and is often taught in introductory algebra courses. It is easy to memorize and can be applied quickly to solve equations.

Completing the square is another method of solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be factored to solve for x. This method is often used in more advanced algebra courses and is particularly useful in solving equations that cannot be factored easily.

One advantage of completing the square is that it can be used to solve any quadratic equation, regardless of its coefficients. In contrast, the quadratic formula can become more complicated when the coefficients of the equation are large or complex.

Another advantage of completing the square is that it can be used to derive the quadratic formula. By completing the square on the quadratic equation ax^2 + bx + c = 0, one can arrive at the quadratic formula.

However, completing the square can be a more time-consuming method of solving quadratic equations compared to the quadratic formula. It requires more steps and can be more difficult to apply to complex equations.

In summary, both the quadratic formula and completing the square are effective methods of solving quadratic equations. The quadratic formula is a straightforward method that is easy to memorize and apply quickly. Completing the square is a more advanced method that can be used to solve any quadratic equation and can be used to derive the quadratic formula.

Solving Quadratic Equations by Completing the Square FAQ

What is the completing the square method for solving quadratic equations?

Completing the square is a method used to solve quadratic equations. It involves manipulating the equation to rewrite it in a different form, which makes it easier to solve. The goal is to create a perfect square trinomial, which can be factored into a binomial squared.

How do you use the completing the square method to solve a quadratic equation?

To use the completing the square method, the quadratic equation must first be in standard form, which is ax^2 + bx + c = 0. Then, follow these steps:

1. Divide both sides of the equation by the coefficient of the x^2 term, so that the coefficient becomes 1.
2. Move the constant term to the right side of the equation.
3. Add and subtract the square of half the coefficient of the x term on the left side of the equation.
4. Factor the perfect square trinomial on the left side of the equation.
5. Take the square root of both sides of the equation.
6. Solve for x.

What is the difference between completing the square and factoring?

Completing the square and factoring are both methods used to solve quadratic equations, but they are different. Completing the square involves manipulating the equation to create a perfect square trinomial, which can then be factored into a binomial squared. Factoring involves finding two numbers that multiply to the constant term and add to the coefficient of the x term, and then using those numbers to write the quadratic equation in factored form.

Can you solve a quadratic equation by completing the square with a coefficient?

Yes, you can solve a quadratic equation by completing the square with a coefficient. The steps are the same as when the coefficient is 1, but you must divide both sides of the equation by the coefficient of the x^2 term before beginning the process.

What are some common mistakes to avoid when using the completing the square method?

Some common mistakes to avoid when using the completing the square method include:

• Forgetting to divide by the coefficient of the x^2 term before beginning the process.
• Making errors when adding and subtracting the square of half the coefficient of the x term.
• Making errors when factoring the perfect square trinomial.
• Forgetting to take the square root of both sides of the equation.
• Making errors when solving for x.

Are there any real-world applications for solving quadratic equations by completing the square?

Yes, there are many real-world applications for solving quadratic equations by completing the square. For example, it can be used to determine the maximum or minimum value of a quadratic function, which is important in optimization problems. It can also be used in physics to determine the trajectory of a projectile.

How do you solve quadratic equations by completing the square?

To solve quadratic equations by completing the square, follow the steps outlined in the previous question. Divide both sides of the equation by the coefficient of the x^2 term, add and subtract the square of half the coefficient of the x term, factor the perfect square trinomial, take the square root of both sides of the equation, and solve for x.