What Is a Quadratic Equation? Standard Form and Examples

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.