Completing the Square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial . Show To solve a x 2 + b x + c = 0 by completing the square: 1. Transform the equation so that the constant term, c , is alone on the right side. 3. Add the square of half the coefficient of the x -term, ( b 2 a ) 2 to both sides of the equation. 4. Factor the left side as the square of a binomial. 5. Take the square root of both sides. (Remember: ( x + q ) 2 = r is equivalent to x + q = ± r .) 6. Solve for x . Example 1: Solve x 2 − 6 x − 3 = 0 by completing the square. x 2 − 6 x = 3 x 2 − 6 x + ( − 3 ) 2 = 3 + 9 ( x − 3 ) 2 = 12 x − 3 = ± 12 = ± 2 3 x = 3 ± 2 3 Example 2: Solve: 7 x 2 − 8 x + 3 = 0 7 x 2 − 8 x = − 3 x 2 − 8 7 x = − 3 7 x 2 − 8 7 x + ( − 4 7 ) 2 = − 3 7 + 16 49 ( x − 4 7 ) 2 = − 5 49 x − 4 7 = ± 5 7 i x = 4 7 ± 5 7 i ( x − 3 ) 2 = 12 x − 3 = ± 12 = ± 2 3 x = 3 ± 2 3 Calculator UseThis calculator is a quadratic equation solver that will solve a second-order polynomial equation in the form ax2 + bx + c = 0 for x, where a ≠ 0, using the completing the square method. The calculator solution will show work to solve a quadratic equation by completing the square to solve the entered equation for real and complex roots. Completing the square when a is not 1To complete the square when a is greater than 1 or less than 1 but not equal to 0, factor out the value of a from all other terms. For example, find the solution by completing the square for: \( 2x^2 - 12x + 7 = 0 \) \( a \ne 1, a = 2 \) so divide through by 2 \( \dfrac{2}{2}x^2 - \dfrac{12}{2}x + \dfrac{7}{2} = \dfrac{0}{2} \) which gives us \( x^2 - 6x + \dfrac{7}{2} = 0 \) Now, continue to solve this quadratic equation by completing the square method. Completing the square when b = 0When you do not have an x term because b is 0, you will have a easier equation to solve and only need to solve for the squared term. For example: Solution by completing the square for: \( x^2 + 0x - 4 = 0 \) Eliminate b term with 0 to get: \( x^2 - 4 = 0 \) Keep \( x \) terms on the left and move the constant to the right side by adding it on both sides \( x^2 = 4\) Take the square root of both sides \( x = \pm \sqrt[]{4} \) therefore \( x = + 2 \) \( x = - 2 \) How to Complete the SquareIn a regular algebra class, completing the square is a very useful tool or method to convert the quadratic equation of the form y = a{x^2} + bx + c also known as the “standard form”, into the form y = a{(x - h)^2} + k which is known as the vertex form. Find the Vertex Form of y = a{x^2} + bx + c using Completing the SquareExample 1: Find the vertex form of the quadratic function below. This quadratic equation is in the form y = a{x^2} + bx + c. However, I need to rewrite it using some algebraic steps in order to make it look like this… This is the vertex form of the quadratic function where \left( {h,k} \right) is the vertex or the “center” of the quadratic function or the parabola. Before I start, I realize that a = 1. Therefore, I can immediately apply the “completing the square” steps. STEP 1: Identify the coefficient of the linear term of the quadratic function. That is the number attached to the x-term. STEP 2: I will take that number, divide it by 2 and square it (or raise to the power 2). STEP 3: The output in step #2 will be added and subtracted on the same side of the equation to keep it balanced. Think About It: If I add 4 on the right side of the equation, then I am technically changing the original meaning of the equation. So to keep it unchanged, I must subtract the same value that I added on the same side of the equation. STEP 4: Now, express the trinomial inside the parenthesis as a square of a binomial, and simplify the outside constants.
Example 2: Find the vertex form of the quadratic function below. The approach to this problem is slightly different because the value of “a” does not equal to 1, a \ne 1. The first step is to factor out the coefficient 2 between the terms with x-variables only. STEP 1: Factor out 2 only to the terms with variable x. STEP 2: Identify the coefficient of the x-term or linear term. STEP 3: Take that number, divide it by 2, and square. STEP 4: Now, I will take the output \large{9 \over 4} and add it inside the parenthesis.
STEP 5: Since I added \large{9 \over 2} to the equation, then I should subtract the entire equation by \large{9 \over 2} also to compensate for it. STEP 6: Finally, express the trinomial inside the parenthesis as the square of binomial and then simplify the outside constants. Be careful combining the fractions.
Example 3: Find the vertex form of the quadratic function below. Solution:
Example 4: Find the vertex form of the quadratic function below. Solution:
Page 1 of 2 | Next Page What is the formula for completing the square?In mathematics, completing the square is used to compute quadratic polynomials. Completing the Square Formula is given as: ax2 + bx + c ⇒ (x + p)2 + constant. The quadratic formula is derived using a method of completing the square. Let's see.
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