Completing the Square
Introduction
Completing the square is a method used in quadratic equations and expressions. This is a method using which we can write a quadratic equation given in standard form in its vertex form. This is done by rearranging the expression in standard form to write it as a perfect square trinomial.
Completing the square is a powerful method for solving quadratic equations and expressing quadratic functions in vertex form. For solving any quadratic equations we need to factorize it and use its factors to find the roots or zeroes or the solutions to the equation. Sometimes it's very complex or impossible to factorize a quadratic equation given in standard form. For example, we can not factorize `x^2 + 4x + 5` because we can’t find two numbers whose sum is `4` and whose product is `5`. In such cases, completing the square method helps us in factorizing by writing the quadratic equation in vertex form.
What Is Completing the Square?
Completing the square is a method used to rewrite a quadratic expression in the form of a perfect square trinomial. This technique is particularly useful when solving quadratic equations, graphing quadratic functions, or converting quadratic expressions into a more manageable form.
This can be done by rearranging the expression obtained after completing the square: `a(x + m)^2 + n`, such that the left side is a perfect square trinomial. The standard form of a quadratic equation is `ax^2 + bx + c = 0`. Completing the square transforms it into the vertex form `a(x - h)^2 + k`, where `(h, k)` represents the coordinates of the vertex.
Here's a more detailed explanation of completing the square:
Objective:
The primary goal of completing the square is to express a quadratic expression of the form `ax^2 + bx + c` as a perfect square trinomial, which can be written in the form `a(x - h)^2 + k`. This form is called the vertex form of a quadratic function, where `(h, k)` represents the coordinates of the vertex.
Method:
The process of how to complete the square involves several steps:
Step `1`. Ensure the leading coefficient is `1`:
If the leading coefficient of the quadratic term `(ax^2)` is not `1`, factor it out. For instance, if the leading coefficient is `a`, divide the entire expression by `a`. This step ensures that the quadratic term has a coefficient of `1`.
Step `2`. Move the constant term to the other side:
Rewrite the quadratic expression by moving the constant term `(c)` to the opposite side of the equation. For example, if the quadratic expression is in the form `ax^2 + bx - c = 0`, move `c` to the other side to obtain `ax^2 + bx = c`.
Step `3`. Complete the square for the quadratic term:
To complete the square, take the coefficient of the linear term `(b)`, divide it by `2`, and then square the result. Add this squared value to both sides of the equation.
The squared value is `\left(\frac{b}{2}\right)^2 = \frac{b^2}{4}`.
Add `\frac{b^2}{4}` to both sides of the equation.
Step `4`. Factor the perfect square trinomial:
Write the quadratic expression as a perfect square trinomial. This trinomial can be factored into `(x + h)^2`, where `h` is half of the coefficient of the linear term (`b`).
Step `5`. Solve for `x`:
Rewrite the equation in the form `(x + h)^2 = k` and solve for `x`.
Take the square root of both sides of the equation and solve for `x`. This yields two possible solutions, considering the positive and negative square roots.
Let's apply the steps to solve an example.
Example: Use the Completing the Square method to solve the quadratic equation `2x^2 + 8x - 10 = 0`.
Solution:
Here's a step-by-step guide to how you complete the square method:
Step `1`: Ensure leading coefficient is `1`:
If the leading coefficient of the quadratic term `(ax^2)` is not `1`, divide all terms by the leading coefficient to make it equal to `1`. For example, if the quadratic expression is `2x^2 + 8x - 10 = 0`, divide all terms by `2` to get `x^2 + 4x - 5 = 0`.
Step `2`: Move constant term to other side:
Rewrite the equation in the form `ax^2 + bx = -c`. Move the constant term `(c)` to the other side of the equation. For instance, for the quadratic equation `x^2 + 4x - 5= 0`, rewrite it as `x^2 + 4x = 5`.
Step `3`: Complete the square for the quadratic term:
Add and subtract `\left(\frac{b}{2}\right)^2` inside the parenthesis of the quadratic expression. This creates a perfect square trinomial. For example, for the equation `x^2 + 4x = 5`, add and subtract `\left(\frac{4}{2}\right)^2 = 4` to get `x^2 + 4x + 4 = 5 + 4`.
Step `4`: Factor the perfect square trinomial:
Factor the perfect square trinomial into the form `(x + h)^2`, where `h` is half the coefficient of the linear term. Using the previous example, `x^2 + 4x + 4` factors into `(x + 2)^2`.
Step `5`: Solve for `x`:
Rewrite the equation with the perfect square trinomial factored. Solve for `x` by taking the square root of both sides of the equation. Don't forget to consider both the positive and negative square roots. For example, for the equation `(x + 2)^2 = 9`, you would solve for `x` by taking the square root of both sides to get `x + 2 = \pm 3`, which gives `x = -2 \pm 3`, resulting in `x = 1` and `x = -5`.
Step `6`: Check Solutions:
Verify the solutions by substituting them back into the original equation to ensure they satisfy the equation.
Example: Solve the quadratic equation `x^2 + 6x - 3 = 0`.
Solution:
Note that `x^2 + 6x - 3 = 0` is not factorable as there are no two numbers that multiply to give `-3` and add to `6`.
Let's apply the Completing the Square method to solve the quadratic equation `x^2 + 6x - 3 = 0`.
`1`. Identify the equation: `x^2 + 6x - 3 = 0`
`2`. Check Leading Coefficient: Coefficient of `x^2` is `1`, so no need to divide.
`3`. Move Constant Term: Rewrite as `x^2 + 6x = 3`
`4`. Complete the Square: Add and subtract `\left(\frac{6}{2}\right)^2 = 9` inside the parenthesis: `x^2 + 6x + 9 = 3 + 9`
`5`. Factor Perfect Square Trinomial: `(x + 3)^2 = 12`
`6`. Solve for `x`: Rewrite as `(x + 3)^2 = 12`, then take square root: `x + 3 = \pm \sqrt{12}`
`7`. Check Solutions: Verify solutions `x = -3 + \sqrt{12}` and `x = -3 - \sqrt{12}` by substituting them back into the original equation.
Completing the square is particularly useful when solving quadratic equations that cannot be easily factored or when finding the vertex form of a quadratic function. Additionally, it allows for the expression of quadratic equations in a form that reveals important information about their graph, such as the vertex and axis of symmetry.
Formula for Completing the Square
Completing the square formula is yet another technique that can be used to find the roots of a quadratic equation given in its standard form `ax^2 + bx + c`. The Complete the Square formula is again a method used to rewrite a quadratic expression in the form of a perfect square trinomial with some additional constant.
Instead of going through the tedious step by step method for completing the square, we can use the following values of `m` and `n` to rewrite `ax^2 + bx + c` as `a(x + m)^2 + n`, where
`m = \frac{b}{2a}`
`n = c - (\frac{b^2}(4a})`
To complete the square in the expression `ax^2 + bx + c`, first find the values for `m` and `n` using the values of `a, b` and `c,` next substitute these values in
`ax^2 + bx + c = a(x + m)^2 + n`
Let's illustrate the completing the square formula with an example.
Example: Use completing the square formula to solve `x^2 - 4x - 7 = 0`.
Solution:
Here `a = 1, b = -4, c = -7`
Using these values we can find the values for `m` and `n`.
`m = \frac{b}{2a} = -\frac{4}{2} = -2`
`n = c - (\frac{b^2}(4a}) = -7 - (\frac{(-4)^2}(4(1)}) = -11`
Hence we can rewrite `x^2 - 4x - 7 = 0` as `(x - 2)^2 - 11 = 0`
\( (x - 2)^2 - 11 = 0 \)
\( (x - 2)^2 = 11 \)
\( x - 2 = \pm \sqrt{11} \)
\( x = 2 \pm \sqrt{11} \)
Hence the two roots are `x = 2 + \sqrt{11}` and `x = 2 - \sqrt{11}`.
Derivation of Completing the Square Formula
The geometric interpretation of completing the square provides a visual understanding of the completing the square method. By breaking down the quadratic expression into a square and a rectangle, and then adjusting the rectangle to form a perfect square, we can see how the method works.
In this geometric interpretation:
`1`. We start with a square of side length ` x `, representing ` x^2 `.
`2`. We also have a rectangle with length ` \frac{b}{a} ` and width ` x `, representing ` \frac{b}{a}x `.
`3`. We divide the rectangle into two equal parts, each with a length of ` \frac{b}{2a} `.
`4`. We attach half of this rectangle to the right side of the square and the other half to the bottom of the square to complete a larger square.
`5`. To complete the combined square shape, we are short of a square of side `\frac{b}{2a}`. Hence the area of the missing square `[(\frac{b}{2a})^2]` should be added to `x^2 + \frac{b}{2a}x`. However, to balance the equation, if we are adding `[(\frac{b}{2a})^2]`, we need to subtract it as well.
`6`. So, the process results in adding ` \left(\frac{b}{2a}\right)^2 ` and subtracting it as well.
Thus to complete the square,
`x^2 + \frac{b}{a}x = x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - (\frac{b}{2a})^2`
`x^2 + \frac{b}{a}x = x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 - \frac{b^2}{4a^2}`
Multiplying and dividing the right side term `(\frac{b}{a})x` by `2`, gives ` x^2 + \frac{2xb}{2a} + (\frac{b}{2a})^2 - \frac{b^2}{4a^2} `
Using the identity `x^2 + 2xy + y^2 = (x + y)^2` , we can write `x^2 + \frac{b}{a}x` as
`x^2 + \frac{b}{2a}x = (x + \frac{b}{2a})^2 - \frac{b^2}{4a^2}`
By substituting this in `ax^2 + bx + c` we get
`ax^2 + bx + c = a((x + \frac{b}{2a})^2 - \frac{b^2}{4a^2} + \frac{c}{a})`
`= a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} + c`
`= a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})`
This is of the form `a(x + m)^2 + n` where
`m = \frac{b}{2a}`
`n = c - (\frac{b^2}(4a})`
This geometric representation aligns with the algebraic method of completing the square. It illustrates how adding and subtracting ` \left(\frac{b}{2a}\right)^2 ` allows us to rewrite the quadratic expression in a form that can be factored into a perfect square trinomial.
The derived formula ` a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a}) ` from this geometric interpretation matches the result obtained through the step-wise algebraic method.
Real-Life Applications
Completing the square, though often taught as a mathematical technique, has several real-life applications beyond mathematics. Here are a few instances where completing the square is utilized:
`1`. Engineering and Physics: Completing the square is extensively used in engineering and physics, particularly in problems involving optimization, such as maximizing or minimizing certain quantities. For instance, in designing bridges or buildings, engineers often need to optimize the shape of components to minimize material usage while maintaining structural integrity. Completing the square helps in finding the vertex of a parabola, which represents the optimal point.
`2`. Computer Graphics and Animation: In computer graphics and animation, completing the square is used to determine the position of objects in space. It helps in creating smooth animations and realistic simulations by finding the optimal path of motion. For example, when programming the movement of characters in a game or designing animation sequences in movies, equations involving completing the square can be employed to achieve desired trajectories.
`3`. Signal Processing: Completing the square is employed in signal processing applications, such as in audio and image compression algorithms. Techniques like least squares fitting, which involves minimizing the squared error between the observed and predicted values, utilize completing the square to find optimal solutions.
`4`. Finance and Economics: Completing the square is applied in finance and economics for various purposes, including risk management, option pricing models, and portfolio optimization. For example, in financial modeling, completing the square can be used to determine the optimal allocation of assets in a portfolio to maximize returns while minimizing risk.
`5`. Robotics and Control Systems: Completing the square is utilized in robotics and control systems to design controllers that regulate the motion of robots or other automated systems. It helps in optimizing control algorithms to achieve precise and efficient movement while minimizing energy consumption or error.
Solved Examples
Example `1`: Solve the quadratic equation ` 2x^2 + 4x - 6 = 0 ` by completing the square.
Solution:
Step `1`: Divide throughout by the coefficient of `x^2`:
\(x^2 + 2x - 3 = 0\)
Step `2`: Move the constant term to the other side of the equation:
\(x^2 + 2x = 3\)
Step `3`: To complete the square, add and subtract `(\frac{2}{2})^2 = 1` inside the parenthesis:
\(x^2 + 2x + 1 - 1 = 3\)
Step `4`: Factor the perfect square trinomial:
\((x + 1)^2 - 1 = 3\)
Step `5`: Move the constant term to the other side:
\((x + 1)^2 = 4\)
Step `6`: Take the square root of both sides:
\(x + 1 = \pm 2\)
Step `7`: Solve for `x`:
\(x = -1 \pm 2\)
\(x = -1 + 2 = 1 \quad \text{or} \quad x = -1 - 2 = -3\)
So, the solutions to the equation are `x = 1` and `x = -3`.
Example `2`: Solve the quadratic equation `3x^2 - 6x + 2 = 0` by completing the square.
Solution:
Step `1`: Divide throughout by the coefficient of `x^2`:
`x^2 - 2x + \frac{2}{3} = 0`
Step `2`: Move the constant term to the other side of the equation:
`x^2 - 2x = -\frac{2}{3}`
Step `3`: To complete the square, add and subtract `(\frac{-2}{2})^2 = 1` inside the parenthesis:
`x^2 - 2x + 1 - 1 = -\frac{2}{3}`
Step `4`: Factor the perfect square trinomial:
`(x - 1)^2 - 1 = -\frac{2}{3}`
Step `5`: Move the constant term to the other side:
`(x - 1)^2 = 1 - \frac{2}{3}`
`(x - 1)^2 = \frac{1}{3}`
Step `6`: Take the square root of both sides:
`x - 1 = \pm \sqrt{\frac{1}{3}}`
Step `7`: Solve for `x`:
`x = 1 \pm \sqrt{\frac{1}{3}}`
So, the solutions to the equation are `x = 1 + \sqrt{\frac{1}{3}}` and `x = 1 - \sqrt{\frac{1}{3}}`.
Example `3`: Use completing the square formula to solve `x^2 + 4x - 5 = 0`.
Solution:
Here `a = 1, b = 4, c = -5`
Using these values we can find the values for `m` and `n`.
`m = \frac{b}{2a} = -\frac{4}{2} = 2`
`n = c - (\frac{b^2}(4a}) = -5 - (\frac{(4)^2}(4(1)}) = -9`
Hence we can rewrite `x^2 + 4x - 5 = 0` as `(x + 2)^2 - 9 = 0`
\( (x + 2)^2 - 9 = 0 \)
\( (x + 2)^2 = 9 \)
\( x + 2 = \pm 3 \)
\( x = -2 \pm 3 \)
So, the solutions to the equation are `x = 1` and `x = -5`.
Example `4`: Solve the quadratic equation `2x^2 - 4x + 3 = 0` by completing the square.
Solution:
Step `1`: Divide throughout by the coefficient of `x^2`:
`x^2 - 2x + \frac{3}{2} = 0`
Step `2`: Move the constant term to the other side of the equation:
`x^2 - 2x = -\frac{3}{2}`
Step `3`: Complete the square by adding and subtracting `(\frac{-2}{2})^2 = 1` inside the parenthesis:
`x^2 - 2x + 1 - 1 = -\frac{3}{2}`
Step `4`: Factor the perfect square trinomial:
`(x - 1)^2 - 1 = -\frac{3}{2}`
Step `5`: Move the constant term to the other side:
`(x - 1)^2 = 1 - \frac{3}{2}`
`(x - 1)^2 = \frac{-1}{2}`
Step `6`: There are no real solutions to the equation since the square of a real number cannot be negative.
So, the equation `2x^2 - 4x + 3 = 0` has no real solutions.
Example `5`: Solve the quadratic equation `3x^2 + 7x - 6 = 0` by completing the square.
Solution:
Step `1`: Divide throughout by the coefficient of `x^2` if it's not `1`:
`x^2 + \frac{7}{3}x - 2 = 0`
Step `2`: Move the constant term to the other side of the equation:
`x^2 + \frac{7}{3}x = 2`
Step `3`: Complete the square by adding and subtracting `(\frac{7}{6})^2 = \frac{49}{36}` inside the parenthesis:
`x^2 + \frac{7}{3}x + \frac{49}{36} - \frac{49}{36} = 2`
Step `4`: Factor the perfect square trinomial:
`\left(x + \frac{7}{6}\right)^2 - \frac{49}{36} = 2`
Step `5`: Move the constant term to the other side:
`\left(x + \frac{7}{6}\right)^2 = 2 + \frac{49}{36}`
`\left(x + \frac{7}{6}\right)^2 = \frac{72}{36} + \frac{49}{36}`
`\left(x + \frac{7}{6}\right)^2 = \frac{121}{36}`
Step `6`: Take the square root of both sides:
`x + \frac{7}{6} = \pm \frac{11}{6}`
Step `7`: Solve for `x`:
`x = -\frac{7}{6} \pm \frac{11}{6}`
`x = -\frac{7}{6} + \frac{11}{6} = \frac{4}{6} = \frac{2}{3}`
`x = -\frac{7}{6} - \frac{11}{6} = -\frac{18}{6} = -3`
So, the solutions to the equation are `x = \frac{2}{3}` and `x = -3`.
Practice Problems
Q`1`. Solve the quadratic equation `x^2 + 6x + 9 = 0` by completing the square.
- `x = -3`
- `x = -3`
- `x = -3`
- `x = -3`
Answer: a
Q`2`. Solve the quadratic equation `4x^2 - 8x + 5 = 0` by completing the square.
- `x = -5`
- No real solutions
- `x = -3`
- `x = 4`
Answer: b
Q`3`. Solve the quadratic equation `x^2 + 10x + 25 = 0` by completing the square.
- `x = 5`
- `x = -4`
- `x = -5`
- No real solutions
Answer: c
Q`4`. Solve the quadratic equation `2x^2 + 4x + 3 = 0` by completing the square.
- `x = -5`
- `x = -4`
- `x = 3`
- No real solutions
Answer: d
Q`5`. Solve the quadratic equation `x^2 - 8x + 15 = 0` by completing the square.
- `x = 5, x = 3`
- `x = -5`
- No real solutions
- `x = 3`
Answer: a
Frequently Asked Questions
Q`1`. What is completing the square?
Answer: Completing the square is a method used in algebra to rewrite quadratic expressions from standard form to vertex form, making it easier to solve or manipulate them.
Q`2`. Why do we complete the square?
Answer: Completing the square helps us solve quadratic equations, find the vertex of a quadratic function, convert quadratic expressions between different forms, and solve problems involving maximum or minimum values.
Q`3`. How do you complete the square?
Answer: To complete the square for a quadratic expression ` ax^2 + bx + c `, follow these steps:
`1`. Group the ` x `-terms together and leave the constant term separate.
`2`. Factor out the leading coefficient ` a ` from the ` x `-terms.
`3`. Add and subtract `(b/2)^2`, which is half the coefficient of the ` x `-term squared.
`4`. Factor the resulting expression as a perfect square trinomial.
`5`. Simplify the expression if possible.
Q`4`. When do we use completing the square?
Answer: Completing the square is particularly useful when solving quadratic equations that are not easily factorable, or when finding the vertex of a quadratic function.
Q`5`. What are the benefits of completing the square?
Answer: Completing the square provides a systematic method to solve quadratic equations and analyze quadratic functions. It helps in finding the vertex form of a quadratic function, which provides information about its vertex and axis of symmetry.
Q`6`. What if the leading coefficient is not `1`?
Answer: If the leading coefficient of the quadratic term is not `1`, you can still complete the square. Begin by dividing all terms by the leading coefficient before following the usual steps. After completing the square, multiply all terms by the leading coefficient to restore the original expression.
Q`7`. Can completing the square be used for any quadratic equation?
Answer: Yes, completing the square is a general method applicable to any quadratic equation, whether it's in standard form or not.
Q`8`. How do you use completing the square to solve quadratic equations?
Answer: To solve a quadratic equation using completing the square, rewrite the equation in the form ` ax^2 + bx + c = 0 `, complete the square to express it as ` (x - h)^2 = k `, and then solve for ` x ` by taking the square root of both sides and isolating ` x `.
Q`9`. What are some common mistakes to avoid when completing the square?
Answer: Common mistakes include forgetting to divide all terms by the leading coefficient if it's not `1`, errors in halving and squaring the coefficient of ` x `, and errors in simplifying the expression after completing the square.