What Is a Quadratic Equation? Standard Form and Examples

A qua­dratic equa­tion is an equa­tion in which the high­est power of the unknown is 2, such as $2x^2 + x - 300 = 0$. Qua­dratic equa­tions turn up when­ever two unknown quan­ti­ties are mul­ti­plied together: the area of a room, the prod­uct of two ages, the total cost when the price depends on the quan­tity, or the time taken when speed changes. This les­son, based on Chap­ter 4 of the NCERT Class 10 text­book, explains what makes an equa­tion qua­dratic, how to write it in stan­dard form, how to turn a word prob­lem into one, and how to test whether an untidy equa­tion is really qua­dratic.

A prob­lem that leads to a qua­dratic equa­tion#

The prayer hall#

A char­ity trust wants to build a prayer hall. The car­pet area must be exactly 300 square metres, and the length of the hall must be one metre more than twice its breadth. What should the length and breadth be?

Give the unknown a let­ter. Let the breadth of the hall be $x$ metres. The length is "one metre more than twice the breadth", so the length is $(2x + 1)$ metres. Now use the area:

$$\text{area} = \text{length} \times \text{breadth} = (2x+1) \times x = 2x^2 + x$$

The area is 300 square metres, so $2x^2 + x = 300$. Mov­ing every term to one side gives

$$2x^2 + x - 300 = 0$$

This equa­tion is dif­fer­ent from the lin­ear equa­tions you already know, such as $3x + 5 = 20$, where $x$ appears only to the first power. Here $x$ is squared, and that changes how the equa­tion must be solved. That is why such equa­tions get their own name and their own chap­ter.

Def­i­n­i­tion and stan­dard form#

What a qua­dratic equa­tion is#

A qua­dratic equa­tion in the vari­able $x$ is an equa­tion of the form

$$ax^2 + bx + c = 0, \qquad a \neq 0$$

where $a$, $b$ and $c$ are real num­bers. The con­di­tion $a \neq 0$ is essen­tial. If $a$ were 0, the $x^2$ term would dis­ap­pear and we would be left with $bx + c = 0$, which is a lin­ear equa­tion.

More gen­er­ally, any equa­tion of the form $p(x) = 0$, where $p(x)$ is a poly­no­mial of degree 2, is a qua­dratic equa­tion.

The names of the parts#

• $a$ is the coef­fi­cient of $x^2$ (also called the lead­ing coef­fi­cient);
• $b$ is the coef­fi­cient of $x$;
• $c$ is the con­stant term.

When the terms are writ­ten in descend­ing order of pow­ers and set equal to zero, the equa­tion is in stan­dard form. The prayer hall equa­tion $2x^2 + x - 300 = 0$ is already in stan­dard form, with $a = 2$, $b = 1$ and $c = -300$.

Recog­nis­ing the shape#

The equa­tions $2x^2 - 3x + 1 = 0$ and $4x - 3x^2 + 2 = 0$ are both qua­dratic. The sec­ond is not writ­ten in stan­dard form, because the squared term is not first. Rear­rang­ing from the high­est power to the low­est gives $-3x^2 + 4x + 2 = 0$, so $a = -3$, $b = 4$ and $c = 2$. If you pre­fer a pos­i­tive lead­ing coef­fi­cient, mul­ti­ply every term by $-1$ to get $3x^2 - 4x - 2 = 0$; this is the same equa­tion, with the same roots.

Why "at most two" mat­ters#

A qua­dratic poly­no­mial has at most two zeroes, so a qua­dratic equa­tion has at most two roots. On a graph, the curve $y = ax^2 + bx + c$ (a parabola) meets the $x$-axis at most twice. You will use this idea again when you study the dis­crim­i­nant.

How to test whether an equa­tion is qua­dratic#

The method, step by step#

1. Expand every bracket on both sides, using iden­ti­ties such as $(a+b)^2 = a^2 + 2ab + b^2$ and $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
2. Bring all terms to one side so that the other side is 0.
3. Col­lect like terms and sim­plify.
4. Look at the high­est power of $x$ that sur­vives. If it is $x^2$ with a non-zero coef­fi­cient, the equa­tion is qua­dratic. If the $x^2$ terms can­cel and only $x$ remains, it is lin­ear. If an $x^3$ term sur­vives, it is cubic.

Four worked checks#

Turn­ing a word prob­lem into a qua­dratic equa­tion#

The method, step by step#

1. Choose a let­ter for one unknown quan­tity and say clearly what it stands for, with units.
2. Write every other unknown quan­tity in terms of that let­ter.
3. Find the con­di­tion in the prob­lem that has not yet been used, and write it as an equa­tion.
4. Expand, bring all terms to one side and write the result in stan­dard form.

Worked exam­ple: mar­bles#

Worked exam­ple: toys#

A short his­tory#

Qua­dratic equa­tions are very old. The Baby­lo­ni­ans are believed to have been the first to solve them: they could find two pos­i­tive num­bers from their sum and their prod­uct, which is the same as solv­ing $x^2 - px + q = 0$. Euclid gave a geo­met­ric way of find­ing lengths that, in today's lan­guage, are roots of qua­dratic equa­tions. Brah­magupta (598–665 CE) gave an explicit for­mula for equa­tions of the form $ax^2 + bx = c$, and Srid­haracharya (about 1025 CE) derived the for­mula now known as the qua­dratic for­mula. The Arab math­e­mati­cian Al-Khwarizmi (about 800 CE) stud­ied qua­dratic equa­tions of dif­fer­ent types, and Abra­ham bar Hiyya Ha-Nasi, in his book Liber embado­rum pub­lished in Europe in 1145 CE, gave com­plete solu­tions of dif­fer­ent qua­dratic equa­tions.

Com­mon mis­takes#

• Decid­ing an equa­tion is qua­dratic because it con­tains $x^2$ some­where, with­out sim­pli­fy­ing. If the $x^2$ terms can­cel, as in Exam­ple 2, it is not qua­dratic.
• Decid­ing an equa­tion is not qua­dratic because it con­tains $x^3$. If the $x^3$ terms can­cel, as in Exam­ple 4, it may well be qua­dratic.
• For­get­ting the con­di­tion $a \neq 0$ in the def­i­n­i­tion.
• Sign errors when expand­ing a prod­uct such as $(x-5)(40-x)$: expand term by term and write every sign.
• Read­ing off $a$, $b$ and $c$ before putting the equa­tion in stan­dard form, for exam­ple tak­ing $a = 4$ in $4x - 3x^2 + 2 = 0$.
• For­get­ting to say what the let­ter stands for, with units, in a word prob­lem.

Try these#

1. Is $(x+3)^2 = x^2 + 6$ a qua­dratic equa­tion? Answer: No. It sim­pli­fies to $6x + 3 = 0$, which is lin­ear.
2. Is $x(x-4) + 2 = (2x-1)(x+1)$ a qua­dratic equa­tion? Answer: Yes. It sim­pli­fies to $x^2 + 5x - 3 = 0$.
3. Write $5 - 2x^2 + 3x = 0$ in stan­dard form and state $a$, $b$ and $c$. Answer: $-2x^2 + 3x + 5 = 0$, so $a = -2$, $b = 3$, $c = 5$ (equiv­a­lently $2x^2 - 3x - 5 = 0$).
4. The prod­uct of two con­sec­u­tive pos­i­tive even inte­gers is 168. Form the qua­dratic equa­tion, tak­ing the smaller inte­ger as $x$. Answer: $x^2 + 2x - 168 = 0$.
5. The sum of a non-zero num­ber and its rec­i­p­ro­cal is $\displaystyle \frac{10}{3}$. Form the qua­dratic equa­tion. Answer: $3x^2 - 10x + 3 = 0$.
6. Is $(x-1)^3 = x^3 + x$ a qua­dratic equa­tion? Answer: Yes. It sim­pli­fies to $-3x^2 + 2x - 1 = 0$, that is, $3x^2 - 2x + 1 = 0$.

Key terms#

Qua­dratic equa­tion

An equa­tion that can be writ­ten as $ax^2 + bx + c = 0$ with $a$, $b$, $c$ real and $a \neq 0$.

Stan­dard form

The arrange­ment $ax^2 + bx + c = 0$, with terms in descend­ing pow­ers of $x$ and zero on the right.

Coef­fi­cient of $x^2$

The num­ber $a$ mul­ti­ply­ing $x^2$; it must not be zero.

Coef­fi­cient of $x$

The num­ber $b$ mul­ti­ply­ing $x$; it may be zero.

Con­stant term

The num­ber $c$ that does not mul­ti­ply any power of $x$.

Degree

The high­est power of the vari­able that remains after the equa­tion is sim­pli­fied.

Root

A value of $x$ that makes the equa­tion true. A qua­dratic equa­tion has at most two roots.

Lin­ear equa­tion

An equa­tion of degree 1, such as $x + 12 = 0$.

Com­mon ques­tions#

Can $b$ or $c$ be zero in a qua­dratic equa­tion?#

Yes. Only $a$ must be non-zero. For exam­ple, $x^2 + 7 = 0$ has $b = 0$, and $2x^2 - 5x = 0$ has $c = 0$; both are qua­dratic.

Is an equa­tion with $x^3$ in it ever qua­dratic?#

It can be, if the $x^3$ terms can­cel when you sim­plify. The equa­tion $(x+2)^3 = x^3 - 4$ reduces to $x^2 + 2x + 2 = 0$, which is qua­dratic.

Why must the lead­ing coef­fi­cient be non-zero?#

If $a = 0$, the $x^2$ term van­ishes and the equa­tion becomes $bx + c = 0$, which is lin­ear. The con­di­tion keeps the squared term present.

Does mul­ti­ply­ing by $-1$ change the equa­tion?#

No. Mul­ti­ply­ing every term by the same non-zero num­ber gives an equiv­a­lent equa­tion with exactly the same roots, so $-x^2 + 45x - 324 = 0$ and $x^2 - 45x + 324 = 0$ have the same solu­tions.

Do I have to solve the equa­tion when a ques­tion says "rep­re­sent the sit­u­a­tion math­e­mat­i­cally"?#

No. Such a ques­tion only asks for the equa­tion in stan­dard form. Solv­ing it is the next step, cov­ered in the les­son on fac­tori­sa­tion.

Ref­er­ences#

1. National Coun­cil of Edu­ca­tional Research and Train­ing. Math­e­mat­ics: Text­book for Class X. NCERT, New Delhi.
2. Sharma, R. D. Math­e­mat­ics for Class 10. Dhan­pat Rai Pub­li­ca­tions.
3. Hall, H. S. and Knight, S. R. Higher Alge­bra. Macmil­lan.