Completing the Square Method

Introduction

Completing the square is an algebraic technique for writing a quadratic expression that contains the perfect square . The Quadratic Formula is the most basic method for determining the roots of a quadratic equation. Certain quadratic equations cannot be easily factorised, and in these cases, we can use this quadratic formula to find the roots as quickly as possible.

The roots of the quadratic equation also aid in determining the sum and product of the roots of the quadratic equation. The quadratic formula's two roots are presented as a single expression. The positive and negative signs can be used alternately to obtain the equation's two distinct roots.

Finding a value (or values) of the variable that satisfies the equation is the first step in solving a quadratic equation. The value(s) that the quadratic equation requires is referred to as its roots, solutions, or zeros. The quadratic equation can only have a maximum of two roots because its degree is 2.

Quadratic Equations

The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$

where a, b, and c are numerical coefficients and x is an unknown variable. Here, ‘a’ is not equal to zero because if it is, the equation will no longer be quadratic and will become linear, such as $\mathrm{bx\:+\:c\:=\:0}$

As a result, we cannot refer to this equation as a quadratic equation.

Another name for the terms a, b, and c is quadratic coefficients. The values of the unknown variable x that fulfill the quadratic equation are the solutions to the problem. Quadratic equations' roots or zeros are known as these solutions. The answers to the given equation are the roots of any polynomial.

Solving Quadratic Equations by Completing the square method

Let the equation is $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$. Then use the steps provided to complete the square technique to answer the problem.

Step 1 − Writing the equation in the form shown will ensure that C is on the right side.

Step 2 − If an is not equal to 1, divide the entire equation by an in such a way that the coefficient of 𝑥² is 1.

Step 3 − Now add $\mathrm{(\frac{b}{2a})^{2}}$ , the square of the term-x coefficient on both sides.

Step 4 − Factorize the binomial term's square on the equation's left side.

Step 5 − Take the square root.

$$\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$$

$$\mathrm{\Longrightarrow\:x^{2}\:+\:\frac{b}{a}x\:+\:\frac{c}{a}\:=\:0}$$

$$\mathrm{\Longrightarrow\:x^{2}\:+\:\frac{b}{a}x\:+\:(\frac{b}{2a})^{2}\:-\:(\frac{b}{2a})^{2}\:+\:\frac{c}{d}\:=\:0}$$

$$\mathrm{\Longrightarrow\:[x\:+\:\frac{b}{2a}]^{2}\:-\:[\frac{b^{2}\:-\:4ac}{4a^{2}}]\:=\:0}$$

$$\mathrm{\Longrightarrow\:[x\:+\:\frac{b}{2a}]^{2}\:=\:[\frac{b^{2}\:-\:4ac}{4a^{2}}]}$$

If $\mathrm{b^{2}\:-\:4ac}$ is greater than or equal to zero then,

$$\mathrm{x\:+\:\frac{b}{2a}\:=\:\pm\:\sqrt{[\frac{b^{2}\:-\:4ac}{4a^{2}}]}}$$

$$\mathrm{\Longrightarrow\:x\:+\:\frac{b}{2a}\:=\:\pm\:\frac{\sqrt{b^{2}\:-\:4ac}}{2a}}$$

$$\mathrm{\Longrightarrow\:x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$$

Quadratic Formula

The roots of a quadratic equation are found using the quadratic formula. In place of the factorization method, this formula aids in evaluating the quadratic equations' solutions. We are aware that the quadratic equation $\mathrm{ax^{2}\:+\:bx}$ has the following solution (or root) formula −

$$\mathrm{x\:+\:\frac{b}{2a}\:=\:\pm\:\frac{\sqrt{b^{2}\:-\:4ac}}{2a}}$$

$$\mathrm{\Longrightarrow\:x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$$

Conditions for No real roots of a Quadratic Equation

A quadratic function has no real roots and the parabola it represents does not intersect the x-axis if its discriminant is less than zero.

• Discriminant, $\mathrm{b^{2}\:-\:4ac\:<0}$

• Parabola does not intersect the x-axis.

Solved Examples

1) Find the root of the quadratic equation $\mathrm{x^{2}\:+\:x\:+\:1\:=\:0}$

Answer − We know that the quadratic formula is $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x\:=\:\frac{-1\:\pm\:\sqrt{1^{2}\:-\:4.1.1}}{2.1}}$$

$$\mathrm{x\:=\:\frac{-1\:\pm\:\sqrt{-3}}{2}}$$

$$\mathrm{x\:=\:\frac{-1\:\pm\:i\:\sqrt{3}}{2}}$$

2) Find the roots of $\mathrm{x^{2}\:-\:2x\:+\:1\:=\:0}$

Answer − We know that the quadratic formula is $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x\:=\:\frac{2\:\pm\:\sqrt{2^{2}\:-\:4.1.1}}{2.1}}$$

$$\mathrm{x\:=\:\frac{2}{2}}$$

$$\mathrm{x\:=\:1}$$

3) Find the root of the quadratic equation $\mathrm{x^{2}\:-\:4x\:+\:3\:=\:0}$

Answer − We know that the quadratic formula is $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x\:=\:\frac{4\:\pm\:\sqrt{(-4)^{2}\:-\:4.1.3}}{2.1}}$$

$$\mathrm{x\:=\:\frac{4\:\pm\:\sqrt{16\:-\:12}}{2}}$$

$$\mathrm{x\:=\:\frac{4\:\pm\:2}{2}}$$

$$\mathrm{\Longrightarrow\:x\:=\:3\:and\:x\:=\:1}$$

4) Find the discriminant for $\mathrm{x^{2}\:-\:2x\:+\:3\:=\:0}$

Answer − Discriminant,

$$\mathrm{b^{2}\:-\:4ac\:=\:(-2)^{2}\:-\:4.1.3}$$

$$\mathrm{=\:4\:-\:12}$$

$$\mathrm{=\:-8}$$

5) Find the root of $\mathrm{}$

Answer − Discriminant,

$$\mathrm{b^{2}\:-\:4ac\:=\:(-5)^{2}\:-\:4.2.3}$$

$$\mathrm{=\:25\:-\:24}$$

$$\mathrm{=\:1}$$

6) Find the roots of the given equation $\mathrm{5x^{2}\:-\:4x\:+\:3\:=\:0}$

Answer − We know that the quadratic formula is $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x\:=\:\frac{4\:\pm\:\sqrt{(-4)^{2}\:-\:4.5.3}}{2.1}}$$

$$\mathrm{x\:=\:\frac{4\:\pm\:\sqrt{16\:-\:60}}{2}}$$

$$\mathrm{x\:=\:\frac{4\:\pm\:i\:\sqrt{44}}{2a}}$$

7) Find the nature of the root of the quadratic equation $\mathrm{x^{2}\:-\:x\:+\:3\:=\:0}$

Answer − For finding the nature of the root of the quadratic equation, we have to find its discriminant, that is

$$\mathrm{b^{2}\:-\:4ac\:=\:(-1)^{2}\:-\:4.1.3}$$

$$\mathrm{=\:1\:-\:12}$$

$$\mathrm{=\:-11}$$

Since $\mathrm{b^{2}\:-\:4ac}$ is less than zero so it has no real roots.

8) Find the nature of the root of the quadratic equation $\mathrm{8x^{2}\:-\:6x\:+\:1\:=\:0}$

Answer − For finding the nature of the root of the quadratic equation we have to find its discriminant, that is

$$\mathrm{b^{2}\:-\:4ac\:=\:(-6)^{2}\:-\:4.8.1}$$

$$\mathrm{=\:36\:-\:32}$$

$$\mathrm{=\:4}$$

Since $\mathrm{b^{2}\:-\:4ac}$ is greater than zero so it has two real roots.

9) Find the nature of the root of the quadratic equation $\mathrm{}$

Answer − For finding the nature of the root of the quadratic equation we have to find its discriminant, that is

$$\mathrm{b^{2}\:-\:4ac\:=\:(-2)^{2}\:-\:4.1.9}$$

$$\mathrm{=\:4\:-36}$$

$$\mathrm{=\:-32}$$

Since $\mathrm{b^{2}\:-\:4ac}$ is less than zero so it has no real roots.

10) Find the root of the quadratic equation $\mathrm{}$

Answer − We know that the quadratic formula is $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

Now apply this formula for the given equation,

$$\mathrm{x\:=\:\frac{3\:\pm\:\sqrt{(-3)^{2}\:-\:4.1.5}}{2.1}}$$

$$\mathrm{x\:=\:\frac{3\:\pm\:\sqrt{9\:-\:20}}{2}}$$

$$\mathrm{x\:=\:\frac{3\:\pm\:i\:\sqrt{11}}{2}}$$

Conclusion

The definition of a quadratic as a second-degree polynomial equation demands that at least one squared term must be included. The quadratic equation has the following generic form − $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$

Completing the square is an algebraic method for writing a quadratic expression in such a way that it contains the perfect square. In simple terms, completing the square is the process of taking a quadratic equation of the form $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$ and changing it to write it in the form $\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$ This method is commonly used to solve quadratic equations.

Finding a value (or values) of the variable that satisfies the equation is the first step in solving a quadratic equation. The value(s) that the quadratic equation requires is referred to as its roots, solutions, or zeros. The quadratic equation can only have a maximum of two roots because its degree is 2.

FAQs

1. What do you mean by quadratic equation?

A quadratic equation is a second-order algebraic equation in x. (a variable). General form − $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0}$

2. What is the condition for no real root?

Discriminant $\mathrm{b^{2}\:-\:4ac\:<0}$ Parabola does not intersect the x-axis

3. What is a quadratic formula?

$\mathrm{x\:=\:\frac{-b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}$

4. What is the general equation for quadratic equations?

The general equation for the quadratic equation is $\mathrm{ax^{2}\:+\:bx\:+\:c\:=\:0\:.\:a\neq\:0}$

5. What is the use of Completing the Square Formula?

By transforming a quadratic polynomial or equation into a perfect square with an additional constant, the square formula is used to factorise it.