### If one root of the quadratic equation is reciprocal of the other, find the value of .

Let $\alpha$ and $\beta$ be the roots of the equation $3 x^{2}-10 x+k=0$. $\therefore \alpha=\frac{1}{\beta} \quad$ (Given) $\Rightarrow \alpha \beta=1$ $\Rightarrow \frac{k}{3}=1 \quad \quad$...

### If the sum of the roots of a quadratic equation is 6 and their product is 6 , the equation is(a) (b) (c) (d)

Answer is (a) $x^{2}-6 x+6=0$ Given: Sum of roots $=6$ Product of roots $=6$ Thus, the equation is: $x^{2}-6 x+6=0$

### If the product of the roots of the equation is then the value of is(a) (b) (c) (d) 12

Answer is (c) 8 It is given that the product of the roots of the equation $x^{2}-3 x+k=10$ is $-2$ The equation can be rewritten as: $x^{2}-3 x+(k-10)=0$ Product of the roots of a quadratic equation...

### The length of the hypotenuse of a right-angled triangle exceeds the length of the base by and exceeds twice the length of the altitude by . Find the length of each side of the triangle.

Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...

### The length of a rectangle is thrice as long as the side of a square. The side of the square is , more than the width of the rectangle. Their areas being equal, find the dimensions.

Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...

### A rectangular filed in long and wide. There is a path of uniform width all around it, having an area of . Find the width of the path

Let the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...

### The speed of a boat in still water is . It can go upstream and downstream is 5 hours. Fid the speed of the stream.

Speed of the boat in still water $=8 \mathrm{~km} / \mathrm{hr}$. Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Speed...

### A train travels at a uniform speed. If the speed had been more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

### The ratio of the sum and product of the roots of the equation is(a) (b) (c) (d)

Answer is (d) $2: 3$ Given: $7 x^{2}-12 x+18=0$ $\therefore \alpha+\beta=\frac{12}{7}$ and $\beta=\frac{18}{7}$, where $\alpha$ and $\beta$ are the roots of the equation $\therefore$ Ratio of the...

### The sum of the roots of the equation is(a) 2(b) (c) 6(d)

Answer is (b) -2 Sum of the roots of the equation $x^{2}-6 x+2=0$ is $\alpha+\beta=\frac{-b}{a}=\frac{-(-6)}{1}=6$, where $\alpha$ and $\beta$ are the roots of the equation.

### Two water taps together can fill a tank in 6 hours. The tap of larger diameter takes 9 hours less than the smaller one to fill the tank separately. Find the time which each tap can separately fill the tank.

Let the tap of smaller diameter fill the tank in $x$ hours. $\therefore$ Time taken by the tap of larger diameter to fill the tank $=(x-9) h$ Suppose the volume of the tank be $V$. Volume of the...

### A train travels at a uniform speed. If the speed had been more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

### Two years ago, man’s age was three times the square of his son’s age. In three years’ time, his age will be four time his son’s age. Find their present ages.

Let son's age 2 years ago be $x$ years. Then, Man's age 2 years ago $=3 x^{2}$ years $\therefore$ Sons present age $=(x+2)$ years Man's present age $=\left(3 x^{2}+2\right)$ years In three years...

### The difference of the squares of two natural numbers is 45 . The square of the smaller number is four times the larger number. Find the numbers.

Let the greater number be $x$ and the smaller number be $y$. According to the question: $\begin{array}{l} x^{2}-y^{2}=45 \\ y^{2}=4 x \end{array}$ From (i) and (ii), we get: $x^{2}-4 x=45$...

### Divide 16 into two parts such that twice the square of the larger part exceeds the square of the smaller part by 164 .

Let the larger and smaller parts be $x$ and $y$, respectively. According to the question: $\begin{array}{l} x+y=16 .....(i) \\ 2 x^{2}=y^{2}+164.....(ii) \end{array} \quad$ From (i), we get:...

### If the roots of the quadratic equation are real and equal, show that either or

$\left(c^{2}-a b\right) x^{2}-2\left(a^{2}-b c\right) x+\left(b^{2}-a c\right)=0$ Here, $a=\left(c^{2}-a b\right), b=-2\left(a^{2}-b c\right), c=\left(b^{2}-a c\right)$ It is given that the roots of...

### If the quadratic equation has equal roots, prove that

$\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ Here, $a=\left(1+m^{2}\right), b=2 m c \text { and } c=\left(c^{2}-a^{2}\right)$ It is given that the roots of the equation are equal;...

### Find the nonzero value of for which the roots of the quadratic equation . are real and equal.

The given equation is $9 x^{2}-3 k x+k=0$. This is of the form $a x^{2}+b x+c=0$, where $a=9, b=-3 k$ and $c=k$. $\therefore D=b^{2}-4 a c=(-3 k)^{2}-4 \times 9 \times k=9 k^{2}-36 k$ The given...

### Find the nature of roots of the following quadratic equations:(i) (ii)

(i) The given equation is $2 x^{2}-8 x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=2, b=-8$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-8)^{2}-4 \times 2 \times 5=64-40=24>0$...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .

The given equation is $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=6$ and $C=-\left(a^{2}+2 a-8\right)$ $\therefore$ Discriminant, $D=$...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:

The given equation is $\begin{array}{l} x+\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}+1}{x}=3 \\ \Rightarrow x^{2}+1=3 x \\ \Rightarrow x^{2}-3 x+1=0 \end{array}$ This equation is of the form...

### Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:

The given equation is $3 x^{2}-2 x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=(-2)^{2}-4 \times 3 \times 2=4-24=-20<0$ Hence,...

### Find the discriminant of the given equation:

Here, $\begin{array}{l} a=2 \\ b=-7 \\ c=6 \end{array}$ Discriminant $D$ is diven by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-7)^{2}-4 \times 2 \times 6 \\ =49-48 \\ =1 \end{array}$

### Find the roots of the given equation:

We write: $2 \sqrt{2} x=3 \sqrt{2} x-\sqrt{2} x$ as $x^{2} \times(-6)=-6 x^{2}=3 \sqrt{2} x \times(-\sqrt{2} x)$ $\therefore x^{2}+2 \sqrt{2} x-6=0$ $\Rightarrow x^{2}+2 \sqrt{2} x-\sqrt{2} x-6=0$...

### Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

$\left( \mathbf{iii} \right)\text{ }\mathbf{4}{{\mathbf{x}}^{\mathbf{2}}}~+~\mathbf{4}\surd \mathbf{3x}~+\text{ }\mathbf{3}\text{ }=\text{ }\mathbf{0}$ On comparing the given equation...

### The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Solution: Let us say, the base of the right triangle be x cm. Given, the altitude of right triangle = (x – 7) cm From Pythagoras theorem, we know, Base2 + Altitude2 = Hypotenuse2 ∴ x2 + (x – 7)2 =...

### Represent the following situations in the form of quadratic equations: (iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age. (ii) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken

(iii) Let us consider, Age of Rohan’s = x  years Therefore, as per the given question, Rohan’s mother’s age = x + 26 After 3 years, Age of Rohan’s = x + 3 Age of Rohan’s mother will be = x + 26 + 3...

Solutions: (i) Let us consider, Breadth of the rectangular plot = x m Thus, the length of the plot = (2x + 1) m. As we know, $Area\text{ }of\text{ }rectangle\text{ }=\text{ }length~\times... read more ### Check whether the following are quadratic equations: (i) (x + 2)3 = 2x (x2 – 1) (ii) x3 – 4×2 – x + 1 = (x – 2)3$\left( vii \right)\text{ }Given,\text{ }{{\left( x~+\text{ }2 \right)}^{3}}~=\text{ }2x({{x}^{2}}~\text{ }1)$By using the formula for${{\left( a+b \right)}^{2~}}=\text{ }{{a}^{2}}+2ab+{{b}^{2}}$... read more ### Check whether the following are quadratic equations: (i) (2x – 1)(x – 3) = (x + 5)(x – 1) (ii) x2 + 3x + 1 = (x – 2)2 (i) Given, (2x – 1)(x – 3) = (x + 5)(x – 1) By using the formula for (a+b)2=a2+2ab+b2 ⇒ 2x2 – 7x + 3 = x2 + 4x – 5 ⇒ x2 – 11x + 8 = 0 Since the above equation is in the form of ax2 + bx + c = 0.... read more ### Check whether the following are quadratic equations: (i) (x – 2)(x + 1) = (x – 1)(x + 3) (ii) (x – 3)(2x +1) = x(x + 5)$\left( iii \right)\text{ }Given,\text{ }\left( x\text{ }\text{ }2 \right)\left( x\text{ }+\text{ }1 \right)\text{ }=\text{ }\left( x\text{ }\text{ }1 \right)\left( x\text{ }+\text{ }3 \right)$By... read more ### Check whether the following are quadratic equations: (i) (x + 1)2 = 2(x – 3) (ii) x2 – 2x = (–2) (3 – x) (i) Given, (x + 1)2 = 2(x – 3) By using the formula for (a+b)2 = a2+2ab+b2 ⇒ x2 + 2x + 1 = 2x – 6 ⇒ x2 + 7 = 0 Since the above equation is in the form of ax2 + bx + c = 0. Therefore, the given... read more ### 1. A takes days less than the time taken by to finish a piece of work. If both and together can finish the work in days, find the time taken by to finish the work. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let’s consider that$B$takes$x$days to complete the piece of work.... read more ### 1. The hypotenuse of a right triangle is cm. The difference between the lengths of the other two sides of the triangle is cm. Find the lengths of these sides. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the length of one side of the right triangle be$x$cm So, the... read more ### 3. A fast train takes one hour less than a slow train for a journey of km. If the speed of the slow train is km/hr less than that of the fast train, find the speed of the two trains. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let’s consider the speed of the fast train as x km/hr Then, the speed... read more ### 2. A train, traveling at a uniform speed for km, would have taken minutes less to travel the same distance if its speed were km/hr more. Find the original speed of the train. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the original speed of train be$x$km/hr When increased by$5$,... read more ### 1. The speed of a boat in still water is km/hr. It can go km upstream and km downstream in hours. Find the speed of the stream. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Let the speed of stream be$x$km/hr Given, speed of boat in still... read more ### Find the roots of the following quadratic equations (if they exist) by the method of completing the square. 4. Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Given equation,$2{{x}^{2}}+x-4=02\left(...

Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution: Given equation, ${{x}^{2}}+\frac{11}{3}+\frac{10}{3}=0$...
Quadratic is a type of problem which deals with a variable multiplied by itself- an operation also known as squaring. Solution:  Given equation, ${{x}^{2}}-4\sqrt{2x}+6=0$ \${{x}^{2}}-2\times x\times...