Completing the Square Calculator – Online Calculator

You can easily solve the quadratic equation: \(x^2 + 6x + 8 = 0\) and find the value of x without using completing the Square Calculator. But what would be the situation when imaginary numbers come or the equation does not provide any real root?

Don’t worry! Here is your completing the Square Calculator by which solving quadratic equations will be a magic. Before using solving quadratic equations by completing the square calculator, you have to be familiar with some facts related to the equation.

First of all, completing the square is a method used in algebra to manipulate a quadratic expression and express it in a specific form. The customary form of a quadratic equation is:

\[ax^2 + bx + c = 0 \quad \text{where} \quad a \neq 0\]

In the quadratic equation, each term and variable has a specific role. Let’s define each part of the quadratic equation:

ax²:

1. a – which is coefficient of the quadratic term.
2. x – which is variable elevated to the power of 2 (quadratic term).

bx:

1. b – which is coefficient of the linear term.
2. x – which is variable elevated to the power of 1 (linear term).

c:

1. c – which is constant term (no variable attached).

From the equation given above, we can find the value of x as-

\[x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\]

To prove this, you can use the following steps:

Given,

\(ax^2 + bx + c = 0 \quad \text{where} \quad a \neq 0\)

\(\Rightarrow ax^2 + bx = -c\) [Move the constant term to the other side]

\( \Rightarrow x^2 + \frac{bx}{2a} + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \quad \text{[Add $\left(\frac{b}{2a}\right)^2$ to both sides]}\)

\( \Rightarrow x^2 + \frac{bx}{2a} + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \quad \text{[Add $\left(\frac{b}{2a}\right)^2$ to both sides]}\)

\( \Rightarrow x^2 + 2 \cdot x \cdot \frac{b}{2a} + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \quad \text{[Apply $(a^2 + 2ab + b^2)$ to the left side]}\)

\( \left(x + \frac{b}{2a}\right)^2 = \left(-\frac{4ac}{4a^2} + \frac{b^2}{4a^2}\right) \quad \text{[Form $(a+b)^2 = a^2 + 2ab + b^2$]} \)
\( x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 – 4ac}{4a^2}} \quad \text{[Square root both sides]} \)
\( x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 – 4ac}}{2a} \)
\( x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 – 4ac}}{2a} \quad \text{[Simplify $x$]} \)
\( x = -\frac{b \pm \sqrt{b^2 – 4ac}}{2a} \)

From the value of x, it may come to mind, why this type of weird equation is needed. The reason why to Use of Completing the Square is as follows:

1. Solving Quadratic Equations: The quadratic formula got from completing the square is a powerful device for finding the solutions to quadratic equations.
2. Graphing Quadratic Functions: The completed square form aids in graphing quadratic functions and detecting crucial features such as the vertex.
3. Understanding the Nature of Solutions: The discriminant (b²−4ac) in the quadratic formula helps fix whether the solutions are real, imaginary, or repeated.

Here the term discriminant defines that it is a crucial factor in defining the nature of the roots of a quadratic equation. It is calculated using the formula b²−4ac. Its value provides information about the type of solutions:

01. Positive Discriminant:

1. Two distinct real roots.
2. The quadratic expression intersects the x-axis at two distinct points.

02. Zero Discriminant:

1. One real root (equal or repeated roots).
2. The quadratic expression touches the x-axis at one point.

03. Negative Discriminant:

1. Two complex (conjugate) roots.
2. The quadratic expression does not intersect the x-axis.

From the definition of discriminant, you will find three types of roots. These are –

1. Real Roots: Real roots are solutions that lie on the real number line. The discriminant (b²−4ac) is positive i.e. b²−4ac>0

Example: x²−4x+4=0

Here, the discriminant is b²−4ac=(−4)²−4(1)(4) = 0. The equation has one real root (x=2) with multiplicity 2.

2. Complex Roots: Complex roots are solutions that involve imaginary numbers. The discriminant is negative i.e. b²−4ac<0

Example: x²+4 = 0

Here, the discriminant is b²−4ac = 0²−4(1)(4)=−16. The roots are complex numbers: x=2i and x=−2i.

3. Equal (or repeated) Roots: Equal roots occur when the discriminant is zero, resulting in a perfect square trinomial. The quadratic equation has a repeated root i.e. b²−4ac=0

Example: x²−6x+9 = 0

Here, the discriminant is b2−4ac=(−6)²−4(1)(9)=0. The equation has one real root (x=3) with multiplicity 2.

Given that manual completion of these calculations can be time-consuming and vulnerable to errors, a Completing the Square Calculator gifts a suitable and competent method to modernize the process. Prepared with a user-friendly interface, this calculator enables swift determination of solutions to quadratic equations and spares individuals, students, and professionals the need to perform each calculation manually.

Guidelines to use the Completing the Square Calculator

\[ {Formula : } ax^2 + bx + c = 0 \quad \text{where} \quad a \neq 0 \]

From this, \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)

For example: \(x^2 + 6x + 8 = 0\)

1. Input values with their signs (if negative) into A, B and C respectively. Here, A = 1, B = 6 and C = 8
2. Click CALCULATE. You will find your answer with step-by-step calculation.

In the Answer section, you will find:

\( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)

\( = \frac{-6 \pm \sqrt{6^2 – 4 \cdot 1 \cdot 8}}{2 \cdot 1} \)

\( = \frac{-6 \pm \sqrt{36 – 32}}{2} \)

\( = \frac{-6 \pm \sqrt{4}}{2} \)

\( = \frac{-6 \pm 2}{2} \)

\( x = (-6 + 2)/2 = -4/2 = -2 \)

\( \text{Or,} \quad x = (-6 – 2)/2 = -8/2 = -4 \)

If any imaginary number encounters, it will be termed as “i”.

To sum up, completing the Square Calculator streamlines the process of solving quadratic equations by providing a user-friendly interface for quick and accurate results. From handling real and imaginary roots to understanding discriminants, this tool simplifies complex calculations and makes it an essential resource for individuals, students, and professionals. The calculator’s efficiency in solving equations and analyzing quadratic functions highlights its significance in mastering mathematical concepts.

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