Quadratic equations and their roots

A lin­ear equa­tion such as $3x + 5 = 11$ has the unknown only to the first power, and it has exactly one solu­tion. When the unknown appears squared, as in $x^2 - 5x + 6 = 0$, the equa­tion is called qua­dratic, and it can have two solu­tions. Qua­dratic equa­tions appear when­ever an area, a prod­uct of two unknowns, or a speed and a time are involved. This les­son sets out what a qua­dratic equa­tion is, how to write it in stan­dard form, and what is meant by its roots.

Stan­dard form#

An equa­tion of the form

$$ax^2 + bx + c = 0$$

where $a$, $b$, $c$ are real num­bers and $a \neq 0$, is called a qua­dratic equa­tion in $x$. This is its stan­dard form. The num­bers $a$, $b$ and $c$ are called the coef­fi­cients: $a$ is the coef­fi­cient of $x^2$, $b$ the coef­fi­cient of $x$, and $c$ the con­stant term.

The con­di­tion $a \neq 0$ mat­ters: if $a = 0$, the $x^2$ term dis­ap­pears and the equa­tion is lin­ear, not qua­dratic. But $b$ or $c$ may be zero.

Bring­ing an equa­tion to stan­dard form#

Many equa­tions are qua­dratic in dis­guise. Expand any brack­ets, clear frac­tions, and col­lect every term on one side.

Exam­ple 1#

Write $x(x - 2) = 5$ in stan­dard form.

Expand: $x^2 - 2x = 5$. Move 5 to the left: $x^2 - 2x - 5 = 0$. So $a = 1$, $b = -2$, $c = -5$.

Exam­ple 2#

Is $(x + 1)^2 = x^2 + 3$ a qua­dratic equa­tion?

Expand the left side: $x^2 + 2x + 1 = x^2 + 3$. The $x^2$ terms can­cel, leav­ing $2x - 2 = 0$, which is lin­ear. So it is not a qua­dratic equa­tion, even though it con­tains $x^2$. Always sim­plify before decid­ing.

Exam­ple 3#

Write $\displaystyle \frac{x}{2} + \frac{3}{x} = 2$ in stan­dard form, where $x \neq 0$.

Mul­ti­ply every term by $2x$, the com­mon denom­i­na­tor: $x^2 + 6 = 4x$. Col­lect terms: $x^2 - 4x + 6 = 0$.

Exam­ple 4#

Is $(x - 2)(x + 3) = (x + 1)(2x - 1)$ qua­dratic?

Left: $x^2 + x - 6$. Right: $2x^2 + x - 1$. So $x^2 + x - 6 = 2x^2 + x - 1$, which gives $0 = x^2 + 5$, that is $x^2 + 5 = 0$. This is qua­dratic, with $a = 1$, $b = 0$, $c = 5$.

Roots of a qua­dratic equa­tion#

A num­ber is called a root (or a solu­tion) of the equa­tion $ax^2 + bx + c = 0$ if putting it in place of $x$ makes the left side equal to 0. To solve the equa­tion means to find all its roots.

A qua­dratic equa­tion has at most two real roots. It may have two dif­fer­ent roots, one repeated root, or no real root at all. We will see how to tell which in a later les­son.

Exam­ple 5#

Check whether $x = 2$ and $x = 3$ are roots of $x^2 - 5x + 6 = 0$. Is $x = 1$ a root?

For $x = 2$: $4 - 10 + 6 = 0$. Yes, 2 is a root.

For $x = 3$: $9 - 15 + 6 = 0$. Yes, 3 is a root.

For $x = 1$: $1 - 5 + 6 = 2$, not 0. So 1 is not a root.

Exam­ple 6#

If $x = 2$ is a root of $x^2 + kx - 10 = 0$, find $k$.

Since 2 is a root, putting $x = 2$ must give 0:

$$4 + 2k - 10 = 0 \implies 2k = 6 \implies k = 3$$

Exam­ple 7#

If $\displaystyle x = -\frac{1}{2}$ is a root of $2x^2 + px - 3 = 0$, find $p$.

$$\displaystyle 2 \times \frac{1}{4} - \frac{p}{2} - 3 = 0 \implies \frac{1}{2} - \frac{p}{2} = 3 \implies 1 - p = 6 \implies p = -5$$

From words to a qua­dratic equa­tion#

Exam­ple 8#

The prod­uct of two con­sec­u­tive nat­ural num­bers is 132. Write a qua­dratic equa­tion for the smaller num­ber.

Let the smaller num­ber be $x$. The next num­ber is $x + 1$. Their prod­uct is $x(x + 1) = 132$, so

$$x^2 + x - 132 = 0$$

You can check that $x = 11$ is a root: $121 + 11 - 132 = 0$. So the num­bers are 11 and 12.

Exam­ple 9#

The length of a rec­tan­gu­lar plot in Vijayawada is 3 m more than its width, and its area is 108 square metres. Write a qua­dratic equa­tion for the width.

Let the width be $x$ m. The length is $x + 3$ m, and the area is $x(x + 3) = 108$, so $x^2 + 3x - 108 = 0$. (We will solve it in the next les­son: the width is 9 m.)

Prac­tice#

1. Write the val­ues of $a$, $b$, $c$ for $5x^2 - 3x + 8 = 0$ and for $7 - x^2 = 0$.
2. Write $(x - 3)(x + 4) = 10$ in stan­dard form.
3. Write $2x^2 = 3(x + 5)$ in stan­dard form.
4. Is $x^2 + 4x = (x + 2)^2$ a qua­dratic equa­tion? Explain.
5. Is $x(2x + 1) = x^2 + 6$ a qua­dratic equa­tion? If so, write it in stan­dard form.
6. Write $\displaystyle x + \frac{1}{x} = 3$ in stan­dard form, where $x \neq 0$.
7. Check whether $x = -1$ and $x = 4$ are roots of $x^2 - 3x - 4 = 0$.
8. If $x = 3$ is a root of $x^2 - 7x + k = 0$, find $k$.
9. If $\displaystyle x = \frac{2}{3}$ is a root of $3x^2 + mx + 2 = 0$, find $m$.
10. The sum of a num­ber and its square is 90. Write a qua­dratic equa­tion for the num­ber.

Answers#