Word Problems Using Quadratic Formula

We will discuss here how to solve the word problems using quadratic formula.

We know the roots of the quadratic equation ax\(^{2}\) + bx + c = 0, where a ≠ 0 can be obtained by using the quadratic formula x = \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).


1. A line segment AB is 8 cm in length. AB is produced to P such that BP\(^{2}\) = AB  AP. Find the length of BP.

Solution:

Let BP = x cm. Then AP = AB + BP = (8 + x) cm.

Therefore, BP\(^{2}\) = AB ∙ AP

⟹ x\(^{2}\) = 8 ∙ (8 + x)

⟹ x\(^{2}\) - 8x - 64 = 0

Therefore, x = \(\frac{-(-8) \pm \sqrt{(-8)^{2} - 4\cdot 1\cdot (-64)}}{2}\)

x = \(\frac{-8 \pm \sqrt{64 × 5}}{2}\) = \(\frac{-8 \pm 8\sqrt{5}}{2}\)

Therefore, x = 4 ± 4√5.

But the length of BP is positive.

So, x = (4 + 4√5) cm = 4(√5 + 1) cm.


2. In the Annual Sports Meet in a girls’ school, the girls present in the meet, when arranged in a solid square has 16 girls less in the front row, than when arranged in a hollow square 4 deep. Find the number of girls present in the Sports Meet.

Solution:

Let the number of girls in the front row when arranged in a hollow square be x.

Therefore, total number of girls = x\(^{2}\) - (x - 2 × 4)\(^{2}\)

= x\(^{2}\)  - (x - 8)\(^{2}\)

Now, total number of girls when arranged in Solid Square

= (x - 16)\(^{2}\)

According to the condition of the problem,

x\(^{2}\) - (x - 8)\(^{2}\) = (x - 16)\(^{2}\)

⟹ x\(^{2}\) - x\(^{2}\) + 16x - 64 = x\(^{2}\) - 32x + 256

⟹ -x\(^{2}\) + 48x - 320 = 0

⟹ x\(^{2}\) - 48x + 320 = 0

⟹ x\(^{2}\) - 40x - 8x + 320 = 0

⟹ (x - 40)(x - 8) = 0

x = 40 or, 8

But x = 8 is absurd, because the number of girls in the front row of a hollow square 4 deep, must be greater than 8,

Therefore, x = 40

Number of girl students present in the Sports Meet

= (x - 16)\(^{2}\)

= (40 - 16)\(^{2}\)

= 24\(^{2}\)

= 576

Therefore, the required number of girl students = 576


3. A boat can cover 10 km up the stream and 5 km down the stream in 6 hours. If the speed of the stream is 1.5 km/h, find the speed of the boat in still water.

Solution:

Let the speed of the boat in still water be x km/hour.

Then, the speed of the boat up the stream (or against the stream) = (x - \(\frac{3}{2}\)) km/hour, and the speed of the boat down the stream (or along the stream) = (x + \(\frac{3}{2}\)) km/hour.

Therefore, time taken to travel 10 km up the stream = \(\frac{10}{x - \frac{3}{2}}\) hours and time taken to travel 5 km down the stream = \(\frac{5}{x + \frac{3}{2}}\) hours.

Therefore, from the question,

\(\frac{10}{x - \frac{3}{2}}\) + \(\frac{5}{x + \frac{3}{2}}\) = 6

⟹ \(\frac{20}{2x - 3}\) + \(\frac{10}{2x + 3}\) = 6

⟹ \(\frac{10}{2x - 3}\) + \(\frac{5}{2x + 3}\) = 3

⟹ \(\frac{10(2x + 3) + 5(2x – 3)}{(2x – 3)(2x + 3)}\) = 3

⟹ \(\frac{30x + 15}{4x^{2} - 9}\) = 3

⟹ \(\frac{10x + 5}{4x^{2} - 9}\) = 1

⟹ 10x + 5 = 4x\(^{2}\) – 9

⟹ 4x\(^{2}\) – 10x – 14 = 0

⟹ 2x\(^{2}\) -5x – 7 = 0

⟹ 2x\(^{2}\) - 7x + 2x - 7= 0

⟹ x(2x - 7) + 1(2x - 7) = 0

⟹ (2x - 7)(x + 1) = 0

⟹ 2x - 7 = 0 or x + 1 = 0

⟹ x = \(\frac{7}{2}\) or x = -1

But speed cannot be negative. So, x = \(\frac{7}{2}\) = 3.5

Therefore, the speed of the board in still water is 3.5 km/h.


Quadratic Equation

Introduction to Quadratic Equation

Formation of Quadratic Equation in One Variable

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

Examine the Roots of a Quadratic Equation

Problems on Quadratic Equations

Quadratic Equations by Factoring

Word Problems Using Quadratic Formula

Examples on Quadratic Equations 

Word Problems on Quadratic Equations by Factoring

Worksheet on Formation of Quadratic Equation in One Variable

Worksheet on Quadratic Formula

Worksheet on Nature of the Roots of a Quadratic Equation

Worksheet on Word Problems on Quadratic Equations by Factoring




9th Grade Math

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