A mathematical statement in which two expressions on both the left-hand side and the right-hand side of an equality symbol are equal is an equation. Algebra makes it easier to solve real-world situations. Formation of Equations can be done using variables, constant and an equality sign. To use algebra to solve a problem, first convert the problem’s language into mathematical statements that define the connections between the provided data and the unknowns.

Usually, the most challenging part of the procedure is translating the information into mathematical statements. In this article, we will learn how to frame such mathematical statements and how equations can be formed. Continue reading to know more.

Equation and its Types

A mathematical statement containing an ‘equal to’ sign between two algebraic expressions with equal values is known as an equation.

For example, \(2y + 4 = 24\) is an equation. In this,  \(2y + 4\) is the expression on the left-hand side that is equated with the expression \(24\) on the right-hand side.

The degree of an equation is the highest power of variables in a term in an equation. Based on the degree of the equation, equations can be classified as

1. Linear equation, whose degree is \(1\)
2. Quadratic equation, whose degree is \(2\)
3. Cubic equation, whose degree is \(3\)

According to real-life or mathematical situations, each of these equations is framed. When only one unknown is there, then linear equations will help solve it. When a multiplication or squaring of the unknowns is involved, like finding the area, quadratic equations are framed to resolve it. Similarly, when situations demand cubing of the unknowns, like finding the volume of something, we frame the cubic equations. Let us learn to frame each of these equations in detail.

Formation of Linear Equation

An equation whose highest power of the variable or the equation degree is \(1\) is a linear equation. The standard form of a linear equation is \(ax + b\), where \(a,\,b\) are constants and \(a \ne 0\). \(a,\,b\) are respectively the coefficients of \({x^1},\,{x^0}\).

Relationship between Roots and Coefficients of a Linear Equation

If \(\alpha \) is the roots of a quadratic equation \(ax + b\), then

The relationship between the roots and coefficient of a linear equation

\( = – \frac{{{\rm{Constant}}\,{\rm{Term}}}}{{{\rm{Coefficient}}\,{\rm{of}}\,x}} =\, – \frac{b}{a}\)

Obtain a Linear Equation whose Root is Given

A linear equation whose roots are \(\alpha \) is

\((x – \alpha ) = 0\)

\(x = \alpha \)

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For example: Form a linear equation whose root is \(9\).

Let the linear equation be \(ax + b = 0\) and the root is \(\alpha \).

Then, \(\alpha = 9\)

Therefore, the linear equation is given by \(x – 9 = 0\) or \(x = 9\)

Also, by factor theorem, \((x – \alpha )\) is a factor of \(f(x)\) if and only if \(\alpha \) is a root of the quadratic equation \(f(x)\).

Therefore, a quadratic equation \(f(x) = 0\) has:

1. either two distinct real roots
2. two equal real roots
3. no real roots

If a quadratic equation has real roots, it can only have two of them. As a result, a quadratic equation can have no more than two real roots.

With a term with the variable increased to the second power, quadratic equations vary from linear equations. Because adding, subtracting, multiplying, and dividing terms will not isolate the variable, we utilize different strategies to solve quadratic equations than linear equations.

Relation between Roots and the Coefficients of a Quadratic Equation

If \(\alpha ,\,\beta \) are roots of the quadratic equation \(a{x^2} + bx + c = 0\), where \(a \ne 0\), then:

The sum of roots \( = \alpha + \beta = \,- \frac{{{\rm{Coefficient}}\,{\rm{of}}\,x}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^2}}} = \,- \frac{b}{a}\)

The product of roots \( = \alpha \beta = \frac{{{\rm{Constant}}\,{\rm{Term}}}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^2}}} = \frac{c}{a}\)

Obtain a Quadratic Equation whose Roots are Given

A quadratic equation whose roots are \(\alpha \) and \(\beta \) is:

\((x – \alpha )(x – \beta ) = 0\)

\({x^2} – (\alpha + \beta )x + \alpha \beta = 0\)

\({x^2} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}})x + {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

For example: Find a quadratic equation with the sum and the product of its roots as \(4,\,8\) respectively.

Let the equation be \(a{x^2} + bx + c = 0\) and the roots be \(\alpha ,\,\beta \)

Then, the sum of roots \(\alpha + \beta = 4\) 

The product of roots \(\alpha \beta = 8\)

Therefore, a quadratic equation is given by

\({x^2} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}})x + {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

\({x^2} – 4x + 8 = 0\)

So, one quadratic equation which satisfies the given conditions will be \({x^2} – 4x + 8 = 0\).

Formation of a Cubic Equation

An equation whose highest power of the variable or the equation degree is \(3\) is a cubic equation. The standard form of writing a cubic equation is \(a{x^3} + b{x^2} + cx + d\), where \(a,\,b,\,c,\,d\) are constants and \(a \ne 0\).

\(a,\,b,\,c,\,d\) are respectively the coefficients of \({x^3},\,{x^2},\,{x^1},\,{x^0}\).

Relationship between Roots and Coefficients of a Cubic Equation

If \(\alpha ,\,\beta \) and \(\gamma \) are the roots of a cubic equation \(a{x^3} + b{x^2} + cx + d\), then:

The sum of roots \( = \alpha + \beta + \gamma = \,- \frac{{{\rm{Coefficient}}\,{\rm{of}}\,{x^2}}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^3}}} = \,- \frac{b}{a}\)

The sum of the product of roots \( = \alpha \beta + \beta \gamma + \gamma \alpha = \frac{{{\rm{Coefficient}}\,{\rm{of}}\,x}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^3}}} = \frac{c}{a}\)

The product of roots \( = \alpha \beta \gamma = \,- \frac{{{\rm{Constant}}\,{\rm{Term}}}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^3}}} = \,- \frac{d}{a}\)

Obtain a Cubic Equation whose Roots are Given:

A cubic equation whose roots are \(\alpha ,\,\beta \) and \(\gamma \) is:

\((x – \alpha )(x – \beta )(x – \gamma ) = 0\)

\(\left( {{x^2} – \beta x – \alpha x + \alpha \beta } \right)(x – \gamma ) = 0\)

\({x^3} – \beta {x^2} – \alpha {x^2} + \alpha \beta x – \gamma {x^2} + \beta \gamma x + \alpha \gamma x – \alpha \beta \gamma = 0\)

\({x^3} – (\alpha + \beta + \gamma ){x^2} + (\alpha \beta + \beta \gamma + \gamma \alpha )x – \alpha \beta \gamma = 0\)

\({x^3} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}}){x^2} + ({\rm{the}}\,{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{product}}\,{\rm{of}}\,{\rm{root}})x – {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

For example: Find a cubic equation with the sum, sum of the product of its roots taken two at a time, and the product of its roots as \(2, – 7,\, – 14\) respectively.

Let the equation be \(a{x^3} + b{x^2} + cx + d\) and the roots be \(\alpha ,\,\beta \) and \(\gamma \).

Then, the sum of roots \(\alpha + \beta + \gamma = 2\) 

The sum of the product of roots \(\alpha \beta + \beta \gamma + \gamma \alpha = – 7\)

The product of roots \(\alpha \beta \gamma = \,- 14\)

We know that the cubic equation is given by:

\({x^3} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}}){x^2} + ({\rm{the}}\,{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{product}}\,{\rm{of}}\,{\rm{root}})x – {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

\({x^3} – 2{x^2} – 7x + 14 = 0\)

So, one cubic equation which satisfies the given conditions will be \({x^3} – 2{x^2} – 7x + 14 = 0\).

Solved Examples: Formation of Equations

Q.1. If one root of the equation \({x^2} + 12x – \lambda = 0\) is thrice the other. Find the value of \(\lambda \).
Ans: Let \(\alpha ,\,\beta \) are the roots of the given quadratic equation.
According to the given condition, we get:
\(\alpha = 3\beta \)
We know that \(\alpha + \beta = \,- \frac{{12}}{1} = \,- 12\)
\(3\beta + \beta = \,- 12\)
\(4\beta =\, – 12\)
\(\beta = \,- \frac{{12}}{4} = \,- 3\)
\(\alpha = 3\beta = 3 \times ( – 3) =\, – 9\)
Also \(\alpha \beta = \frac{{ – \lambda }}{1}\)
\( – 9 \times ( – 3) = \,- \lambda \)
\(\lambda =\, – 27\)
Hence, the value of \(\lambda \) is \( – 27\)

Q.2. If the product of the roots of \({x^2} – 3x + k = 10\) is \(-2\), what is the value of \(k\)?
Ans: Let \(\alpha ,\,\beta \) are the roots of the given quadratic equation.
According to the given condition, we get:
\(\alpha \beta = \,- 2\)
We know that,
The product of roots \( = \alpha \beta = \frac{{{\rm{Constant}}\,{\rm{term}}}}{{{\rm{Coefficient}}\,{\rm{of}}\,{x^3}}}\)
\( – 2 = \frac{k}{1}\)
\(k = \,- 2\)
Hence, the value of \(k\) is \(-2\).

Q.3. The product of two consecutive even integers is \(168\). Form the quadratic equation to find the integers if \(x\) denotes the smaller integer. Also, find the integers.
Ans: Let the first even integer be \(x\).
So the second consecutive even integer will be \(x + 2\).
Since the given product is \(168\), the equation will be as follows:
\(x(x + 2) = 168\)
\({x^2} + 2x = 168\)
\({x^2} + 2x – 168 = 0\)
Now find the discriminant \(D = {b^2} – 4ac\)
\(D = 4 + 4 \times 168 = 676\)
Since \(D > 0\) two real and distinct roots exist
\(x = \frac{{ – b \pm \sqrt D }}{{2a}}\)
\(x = \frac{{ – 2 \pm \sqrt {676} }}{2}\)
\(x = \frac{{ – 2 \pm 26}}{2}\)
Both roots satisfy the condition of being even integers.
So, the first possibility is two consecutive positive integers are \(12\) and \(14\).
The second possibility is two consecutive negative integers are \(-12\) and \(-14\).

Q.4. Find a cubic equation with the sum, sum of the product of its roots taken two at a time, and the product of its roots as \(3,\, – 5,\,4\) respectively.
Ans: Let the equation be \(a{x^3} + b{x^2} + cx + d = 0\) and the roots be \(\alpha ,\,\beta \) and \(\gamma \).
Then, the sum of roots \(\alpha + \beta + \gamma = 3\) 
The sum of the product of roots \(\alpha \beta + \beta \gamma + \gamma \alpha = – 5\)
The product of roots \(\alpha \beta \gamma = 4\)
We know that the cubic equation is given by:
\({x^3} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}}){x^2} + ({\rm{the}}\,{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{product}}\,{\rm{of}}\,{\rm{root}})x – {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)
\({x^3} – 3{x^2} – 5x – 4 = 0\)
So, one cubic equation which satisfies the given conditions will be \({x^3} – 3{x^2} – 5x – 4 = 0\).

Q.5. Find a linear equation whose root is \(7.\)
Ans: Let the linear equation be \(ax + b = 0\) and the roots are.
Here, \(\alpha = 7\) 
Therefore, the linear equation is given by \(x – 7 = 0\) or \(x = 7\).
So, the linear equation which satisfies the given conditions will be \(x – 7 = 0\).

Summary

This article has discussed an equation and its three types: Linear, Quadratic, and Cubic. Also, we discussed the relationship between the roots and coefficients of linear equations, quadratic equations, and cubic equations. And we discussed in detail how to obtain a linear equation, a quadratic equation, and a cubic equation when the roots are given along with the solved examples.

Frequently Asked Questions

Q.1: How do you make a quadratic equation?
Ans: A quadratic equation can be formed when its roots are given or the sum and product of the roots are given. A quadratic equation whose roots are \(\alpha \) and \(\beta \) is:
\((x – \alpha )(x – \beta ) = 0\) or,
\({x^2} – (\alpha + \beta )x + \alpha \beta = 0\) or,
\({x^2} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}})x + {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

Q.2: How do you make a cubic equation?
Ans: A cubic equation whose roots are \(\alpha ,\,\beta \) and \(\gamma \) are:
\((x – \alpha )(x – \beta )(x – \gamma ) = 0\) or,
\(\left( {{x^2} – \beta x – \alpha x + \alpha \beta } \right)(x – \gamma ) = 0\) or,
\({x^3} – \beta {x^2} – – \alpha {x^2} + \alpha \beta x – \gamma {x^2} + \beta \gamma x + \alpha \gamma x – \alpha \beta \gamma = 0\) or,
\({x^3} – (\alpha + \beta + \gamma ){x^2} + (\alpha \beta + \beta \gamma + \gamma \alpha )x – \alpha \beta \gamma = 0\) or,
\({x^3} – ({\rm{sum}}\,{\rm{of}}\,{\rm{roots}}){x^2} + ({\rm{the}}\,{\rm{sum}}\,{\rm{of}}\,{\rm{the}}\,{\rm{product}}\,{\rm{of}}\,{\rm{root}})x – {\rm{product}}\,{\rm{of}}\,{\rm{roots}} = 0\)

Q.3: What are the types of equations?
Ans: The degree of an equation is the highest power of variables in a term in an equation. Based on the degree of the equation, equations can be classified as:
1. Linear equation, whose degree is \(1\)
2. Quadratic equation, whose degree is \(2\)
3. Cubic equation, whose degree is \(2\)

Q.4: What is an equation?
Ans: A mathematical statement in which two expressions on both the left-hand side and the right-hand side of an equality symbol are equal is an equation.

Q.5: Is x=0 a linear equation?
Ans: Yes, \(x = 0\) is a linear equation as the power of the variable \(x\) is \(1\).

Learn about Quadratic Equations here

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