Quadratic Equations Chapter

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Questions and Answers

What is the general form of a quadratic equation?

x^2 + bx + c = 0

ax + b = 0

ax^2 + bx + c = 0 (correct)

ax^2 + c = 0

In the context of John's and Jivanti's marble problem, what is the final quadratic equation derived?

-x^2 + 45x - 324 = 0

x^2 - 45x + 324 = 0 (correct)

x^2 - 45x + 200 = 0

x^2 - 40x + 324 = 0

In the example of John's marbles, if John lost 5 marbles, how is the number of marbles he has left represented?

x + 5

x - 5 (correct)

45 - x

x - 10

If the total cost of production for toys was ₹750, how is the cost per toy determined in relation to the number of toys produced?

Cost per toy decreases with more toys produced. Signup and view all the answers

What condition is required for an equation to be classified as a quadratic equation?

The polynomial must be of degree 2. Signup and view all the answers

Which of the following statements about quadratic equations is true?

They can arise in various real-life situations. Signup and view all the answers

If the quadratic equation is represented as p(x) = 0, what is true about the degree of p(x)?

It should be exactly 2. Signup and view all the answers

In the equation 2x^2 + x - 300 = 0, what is the coefficient of x^2?

2 Signup and view all the answers

What is the first step in factorising the equation 2x² - 5x + 3 = 0?

Splitting the middle term Signup and view all the answers

What product do the terms -2x and -3x yield when discussing 2x² - 5x + 3?

6x² Signup and view all the answers

For the equation 2x² - 5x + 3 = 0, which of the following represents the roots after factorization?

x = 3/2, x = 1 Signup and view all the answers

What is the correct factorization of the quadratic equation 6x² - x - 2 = 0?

(3x - 2)(2x + 1) Signup and view all the answers

Which statement about the roots of the equation 3x² - 26x + 2 = 0 is true?

The roots are equal and repeated. Signup and view all the answers

When factorising 3x² - 26x + 2, which method is employed?

Grouping the terms Signup and view all the answers

How do you verify that 1 and 3/2 are the roots of 2x² - 5x + 3 = 0?

Substituting them back into the original equation Signup and view all the answers

Which of the following equations can be factorized with roots x = 2/3 and x = 1?

2x² - 5x + 3 Signup and view all the answers

What is the form of the quadratic equation derived from the total cost of production?

x^2 - 55x + 750 = 0 Signup and view all the answers

From the equation (x - 2)² + 1 = 2x - 3, what simplification leads to the conclusion about its quadratic nature?

x^2 - 4x + 5 = 2x - 3 Signup and view all the answers

Which equation is confirmed to not represent a quadratic equation?

x^2 + x + 8 = x^2 - 4 Signup and view all the answers

What is the outcome of the simplification of x(2x + 3) = x^2 + 1?

x^2 + 3x - 1 = 0 Signup and view all the answers

What degree does the equation (x + 2)³ = x³ - 4 represent?

Degree 3 Signup and view all the answers

Which statement correctly describes the equation x^2 + 6x + 8 = 0?

It can be factored easily. Signup and view all the answers

What is the correct form of the equation when simplifying (x + 2)³ = x³ - 4?

6x^2 + 12x + 12 = 0 Signup and view all the answers

Which of these quadratic equations is derived correctly from a provided form?

x^2 - 4x + 4 = 0 Signup and view all the answers

Determine the nature of the roots for the quadratic equation $2x^2 - 3x + 5 = 0$. What is the discriminant value?

-1 Signup and view all the answers

Which condition determines if a quadratic equation has two equal roots?

$b^2 - 4ac = 0$ Signup and view all the answers

For the quadratic equation $2x^2 + kx + 3 = 0$, which of the following values of $k$ will give two equal roots?

-6 Signup and view all the answers

If two friends' ages sum up to 20 years, and four years ago their age product was 48, how old is the younger friend currently?

8 years Signup and view all the answers

Which of the following equations represents a quadratic relation?

$x^2 - 2x + 1 = 0$ Signup and view all the answers

Is it possible to create a rectangular mango grove with a length twice its breadth and an area of 800 m²?

Yes, with length 40 m and breadth 20 m Signup and view all the answers

How many distinct real roots does the quadratic equation $3x^2 - 4x + 4 = 0$ have?

No real roots Signup and view all the answers

What is the perimeter of a rectangular park with an area of 400 m² and breadth of 10 m?

80 m Signup and view all the answers

What is the distance of the pole from gate B when placing it on the boundary of the park?

5 m Signup and view all the answers

What is the discriminant of the equation $x^2 + 7x - 60 = 0$?

289 Signup and view all the answers

What does a positive discriminant indicate about the roots of the quadratic equation?

The equation has two distinct real roots. Signup and view all the answers

According to the Pythagorean theorem applied in this context, what is the relationship expressed as $AP + PB^2 = AB^2$?

$AP + PB^2 = AB^2$ Signup and view all the answers

What values are determined for 'x' after solving the equation $x^2 + 7x - 60 = 0$?

5 and -12 Signup and view all the answers

Why is the distance x = -12 ignored in this context?

Negative distances are not valid. Signup and view all the answers

What is the length of AB, as stated in the problem?

13 m Signup and view all the answers

What does the value of $x$ being positive signify in the context of the problem?

The solution is feasible. Signup and view all the answers

Study Notes

Finding Roots by Factorisation

Factorisation is a method used to solve quadratic equations.
To factorise a quadratic equation, split the middle term into two terms whose product equals the product of the first and last terms.
Once factorised, set each factor equal to zero and solve for x.
The found values of x are the roots of the quadratic equation.

Quadratic Equations: Definition and Standard Form

A quadratic equation in the variable x is of the form ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.
This form is referred to as the standard form.
Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, can be represented as a quadratic equation.

Discriminant and Nature of Roots

The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.
The discriminant determines the nature of the roots:
- b² - 4ac > 0: Two distinct real roots.
- b² - 4ac = 0: Two equal real roots (coincident roots).
- b² - 4ac < 0: No real roots.

Quadratic Formula

The quadratic formula is used to solve quadratic equations and provides the roots of the equation ax² + bx + c = 0.
The formula is: x = (-b ± √(b² - 4ac)) / 2a, where b² - 4ac ≥ 0.

Real-World Applications of Quadratic Equations

Quadratic equations can be applied to various real-life situations involving area, distance, or other quantities represented as quadratic expressions.
Examples include:
- Finding the dimensions of a rectangular garden given its area and length-to-breadth ratio.
- Determining the distance of a pole from a gate in a park using Pythagoras theorem.
- Calculating the age of two friends given the sum and product of their ages at a prior time.

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Description

This quiz explores various aspects of quadratic equations, including factorization methods for finding roots, understanding the standard form, and analyzing the discriminant. Test your knowledge on how to effectively solve and classify quadratic equations based on their characteristics.