The given equation is $x^{2}+6 x+9=0$ Putting $x=-3$ in the given equation, we get $L H S=(-3)^{2}+6 \times(-3)+9=9-18+9=0=R H S$ $\therefore x=-3$ is a solution of the given equation.

### A train covers a distance of at a uniform speed. If the speed had been less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Let the usual speed of the train be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Reduced speed of the train $=(x-8) \mathrm{km} / \mathrm{h}$ Total distance to be covered $=480 \mathrm{~km}$ Time...

### A teacher on attempting to arrange the students for mass drill in the form of solid square found that 24 students were left. When he increased the size of the square by one student, he found that he was short of 25 students. Find the number of students.

Let there be $x$ rows. Then, the number of students in each row will also be $x$. $\therefore$ Total number of students $=\left(x^{2}+24\right)$ According to the question: $\begin{array}{l}...

### The sum of a natural number and its square is Find the number.

Let the required natural number be $x$. According to the given condition, $x+x^{2}=156$ $\Rightarrow x^{2}+x-156=0$ $\Rightarrow x^{2}+13 x-12 x-156=0$ $\Rightarrow x(x+13)-12(x+13)=0$...

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

The given equation is $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{3}, b=-2 \sqrt{2}$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...

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

Given: $x^{2}-6 x+4=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-6$ and $c=4$ Discriminant $D$ is given by: $ \begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 1...

### Find the roots of the given equation:

$\frac{2}{x^{2}}-\frac{5}{x}+2=0$ $\Rightarrow \frac{2-5 x+2 x^{2}}{x^{2}}=0$ $\Rightarrow 2 x^{2}-5 x+2=0$ $\Rightarrow 4 x^{2}-10 x+4=0 \quad$ (Multiplying both sides by 2) $\Rightarrow 4 x^{2}-10...

### Find the roots of the given equation:

$\begin{array}{l} x^{2}-6 x+3=0 \\ \Rightarrow x^{2}-6 x=-3 \\ \Rightarrow x^{2}-2 \times x \times 3+3^{2}=-3+3^{2} \\ \Rightarrow(x-3)^{2}=-3+9=6 \\ \Rightarrow x-3=\pm \sqrt{6} \end{array}$...

### Find the roots of the given equation:

We write, $6 x=(a+4) x-(a-2) x$ as $\begin{array}{l} x^{2} \times\left[-\left(a^{2}+2 a-8\right)\right]=-\left(a^{2}+2 a-8\right) x^{2}=(a+4) x \times[-(a-2) x] \\ \therefore x^{2}+6 x-\left(a^{2}+2...

### Find the roots of the given equation:

$\begin{array}{l} x^{2}-(1+\sqrt{2}) x+\sqrt{2}=0 \\ \Rightarrow x^{2}-x-\sqrt{2} x+\sqrt{2}=0 \\ \Rightarrow x(x-1)-\sqrt{2}(x-1)=0 \\ \Rightarrow(x-\sqrt{2})(x-1)=0 \\ \Rightarrow x-\sqrt{2}=0...

### Find the roots of the given equation:

$\begin{array}{l} 4 \sqrt{6} x^{2}-13 x-2 \sqrt{6}=0 \\ \Rightarrow 4 \sqrt{6} x^{2}-16 x+3 x-2 \sqrt{6}=0 \\ \Rightarrow 4 \sqrt{2} x(\sqrt{3} x-2 \sqrt{2})+\sqrt{3}(\sqrt{3} x-2 \sqrt{2})=0 \\...

### If tanθ + secθ = l, then prove that secθ = (l2 + 1)/2l.

Given: tan θ+ sec θ = l … eq. 1 Duplicating and isolating by (sec θ – tan θ) on numerator and denominator of L.H.S, Along these lines, sec θ – tan θ = 1 … eq.2 Adding eq. 1and eq. 2, we get \[\left(...

### A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan α/(tan β – tan α)].

Considering that an upward banner staff of stature h is overcomed on an upward pinnacle of tallness H(say), with the end goal that FP = h and FO = H. The point of height of the base and top of the...

### A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan α/(tan β – tan α)].

Considering that an upward banner staff of stature h is overcomed on an upward pinnacle of tallness H(say), with the end goal that FP = h and FO = H. The point of height of the base and top of the...

### The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is 30° than when it is 60°. Find the height of the tower.

Let SQ = h be the pinnacle. ∠SPQ = 30° and ∠SRQ = 60° As per the inquiry, the length of shadow is 50 m long hen point of rise of the sun is 30° than when it was 60°. Thus, PR = 50 m and RQ = x m So...

### On the off chance that Zeba were more youthful by 5 years than what she truly is, the square of her age (in years) would have been 11 in excess of multiple times her real age. What is her age now?

Allow Zeba's to age = x As indicated by the inquiry, \[\left( x-5 \right){}^\text{2}=11+5x\] \[x{}^\text{2}+25-10x=11+5x\] \[x{}^\text{2}-15x+14=0\] \[x{}^\text{2}-14x-x+14=0\] \[x\left( x-14...

### A train, going at a uniform speed for 360 km, would have required 48 minutes less to venture to every part of a similar distance if its speed were 5 km/h more. Track down the first speed of the train.

Let unique speed of train = x km/h We know, Time = distance/speed As indicated by the inquiry, we have, Time taken via train = 360/x hour What's more, Time taken via train its speed increment 5 km/h...

### Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

Let the normal number = 'x'. As per the inquiry, We get the condition, \[x{}^\text{2}\text{ }\text{ }84\text{ }=\text{ }3\left( x+8 \right)\] \[x{}^\text{2}\text{ }\text{ }84\text{ }=\text{...

**Find the roots of the quadratic equations by using the quadratic formula in each of the following:**

(½)x2– √11x + 1 = 0 using the formulae: we get,

**Find the roots of the quadratic equations by using the quadratic formula in each of the following:**

x2 + 2 √2x – 6 = 0x2 – 3 √5x + 10 = 0 using formulae: 5. we get, 6.

**Find the roots of the quadratic equations by using the quadratic formula in each of the following:**

–3x2 + 5x + 12 = 0–x2 + 7x – 10 = 0 using the formulae: 3. we get, 4.

**Find the roots of the quadratic equations by using the quadratic formula in each of the following:**

2 x2 – 3x – 5 = 05x2 + 13x + 8 = 0by using ths formulae below: 1. we get, 2.

### A quadratic equation with integral coefficient has integral roots. Justify your answer.

No, a quadratic condition with essential coefficients might possibly have indispensable roots. Support Think about the accompanying condition, \[8x2\text{ }\text{ }2x\text{ }\text{ }1\text{ }=\text{...

**Write whether the following statements are true or false. Justify your answers.**

If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.If the coefficient of x2 and the constant term have the same...

**Write whether the following statements are true or false. Justify your answers.**

Every quadratic equation has at least two roots.Every quadratic equations has at most two roots. (iii) False. For instance, a quadratic condition \[x2\text{ }\text{ }4x\text{ }+\text{ }4\text{...

** Write whether the following statements are true or false. Justify your answers.**

Every quadratic equation has exactly one root.Every quadratic equation has at least one real root. (I) False. For instance, a quadratic condition \[x2\text{ }\text{ }9\text{ }=\text{ }0\] has two...

** State whether the following quadratic equations have two distinct real roots. Justify your answer.**

(x – 1) (x + 2) + 2 = 0(x + 1) (x – 2) + x = 0 (ix) The condition (x – 1) (x + 2) + 2 = 0 has two genuine and unmistakable roots. Working on the above condition, \[x2\text{ }\text{ }x\text{ }+\text{...

**State whether the following quadratic equations have two distinct real roots. Justify your answer.**

√2 x2 –(3/√2)x + 1/√2 = 0x (1 – x) – 2 = 0 (vii) The condition √2x2 – 3x/√2 + ½ = 0 has two genuine and unmistakable roots. \[\begin{array}{*{35}{l}} D\text{ }=\text{ }b2\text{ }\text{...

**State whether the following quadratic equations have two distinct real roots. Justify your answer.**

(x + 4)2 – 8x = 0(x – √2)2 – 2(x + 1) = 0 (v) The condition (x + 4)2 – 8x = 0 has no genuine roots. Working on the above condition, \[x2\text{ }+\text{ }8x\text{ }+\text{...

**State whether the following quadratic equations have two distinct real roots. Justify your answer.**

2x2 – 6x + 9/2 = 03x2 – 4x + 1 = 0 (iii) The condition 2x2 – 6x + (9/2) = 0 has genuine and equivalent roots. \[D\text{ }=\text{ }b2\text{ }\text{ }4ac\] \[=\text{ }\left( -\text{ }6 \right)2\text{...

** State whether the following quadratic equations have two distinct real roots. Justify your answer.**

x2 – 3x + 4 = 02x2 + x – 1 = 0 (I) The condition x2 – 3x + 4 = 0 has no genuine roots. \[D\text{ }=\text{ }b2\text{ }\text{ }4ac\] \[=\text{ }\left( -\text{ }3 \right)2\text{ }\text{...

**Which of the following equations has the sum of its roots as 3?**

(A) 2x2 – 3x + 6 = 0 (B) –x2 + 3x – 3 = 0 (C) √2x2 – 3/√2x+1=0 (D) 3x2 – 3x + 3 = 0 (B) – \[x2\text{ }+\text{ }3x\text{ }\text{ }3\text{ }=\text{ }0\] The amount of the foundations of a...

** Which of the following equations has 2 as a root?**

(A) x2 – 4x + 5 = 0 (B) x2 + 3x – 12 = 0 (C) 2x2 – 7x + 6 = 0 (D) 3x2 – 6x – 2 = 0 (C) \[2x2\text{ }\text{ }7x\text{ }+\text{ }6\text{ }=\text{ }0\] Assuming 2 is a...

**Which of the following is not a quadratic equation?**

(A) 2(x – 1)2 = 4x2 – 2x + 1 (B) 2x – x2 = x2 + 5 (C) (√2x + √3)2 + x2 = 3x2 − 5x (D) (x2 + 2x)2 = x4 + 3 + 4x3 (D) \[\left( x2\text{ }+\text{ }2x...

### 6. Rs. 9000 were divided equally among a certain number of persons. Had there been 20 more persons, each would have got Rs. 160 less. Find the original number of persons

Solution: Let’s consider the original number of people as ‘a’. Given that, $Rs.9000$ were divided equally among a certain number of persons. Had there been $20$ more persons, each would have got...

### 5. If the list price of a toy is reduced by Rs. 2, a person can buy 2 toys more for Rs. 360. Find the original price of the toy.

Solution: Let the original price of the toy be ‘x’. Given that, when the list price of a toy is reduced by $Rs.2$, the person can buy $2$ toys more for $Rs.360$. The number of toys he can buy at the...

### 3. A dealer sells an article for Rs. 24 and gains as much percent as the cost price of the article. Find the cost price of the article.

Solution: Let the cost price be assumed as Rs x. Given, the dealer sells an article for $Rs.24$ and gains as much percent as the cost price of the article. It’s given that he gains as much as the...

### 2. Some students planned a picnic. The budget for food was Rs. 480. But eight of these failed to go and thus the cost of food for each member increased by Rs. 10. How many students attended the picnic?

Solution: Let the number of students who planned the picnic be ‘x’. And given, budget for the food was $Rs.480$ So, cost of food for each member $= 480/x$ Also given, eight of these failed to go and...

### 1. A piece of cloth costs Rs. 35. If the piece were 4 m longer and each metre costs Rs. 1 less, the cost would remain unchanged. How long is the piece?

Solution: Let’s assume the length of the cloth to be ‘a’ meters. Given, piece of cloth costs $Rs.35$ and if the piece were $4m$ longer and each metre costs $Rs.1$ less, the cost remains unchanged....

**Choose the correct answer from the given four options in the following questions: Which of the following is a quadratic equation?**

(A) x2 + 2x + 1 = (4 – x)2 + 3 (B) –2x2 = (5 – x)(2x-(2/5)) (C) (k + 1) x2 + (3/2) x = 7, where k = –1 (D) x3 – x2 = (x – 1)3 (D) \[x3\text{ }\text{ }x2\text{ }=\text{ }\left( x\text{ }\text{ }1...

### 7. An aero plane takes hour less for a journey of if its speed is increased by from its usual speed of the plane. Find its usual speed.

Solution: ...

### 5. The time taken by a person to cover km was more than the time taken in the return journey. If he returned at the speed of more than the speed of going, what was the speed per hour in each direction?

Solution: Let the ongoing speed of person be x km/hr, Then, the returning speed of the person is $= (x + 10) km/hr$ (from the question) Using, speed = distance/ time Time taken by the person in...

### 4. A passenger train takes one hour less for a journey of km if its speed is increased from its usual speed. Find the usual speed of the train.

Solution: Let’s assume the usual speed of train as x km/hr Then, the increased speed of the train $= (x + 5) km/hr$ Using, speed = distance/ time Time taken by the train under usual speed to...

### 3. A fast train takes one hour less than a slow train for a journey of . If the speed of the slow train is less than that of the fast train, find the speed of the two trains

Solution: Let’s consider the speed of the fast train as x km/hr Then, the speed of the slow train will be $= (x -10) km/hr$ Using, speed = distance/ time Time taken by the fast train to cover $200...

### 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.

Solution: Let the original speed of train be x km/hr When increased by $5$, speed of the train $= (x + 5) km/hr$ Using, speed = distance/...

### 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.

Solution: Let the speed of stream be x km/hr Given, speed of boat in still water is $8km/hr$. So, speed of downstream $= (8 + x) km/hr$ And, speed of upstream $= (8 – x) km/hr$ Using, speed =...