Algebra Class 10: Quadratics and Laws

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Questions and Answers

What is the value of the constant term in the expansion of (x + 4)(x - 8)?

-32 (correct)

-12

What is the correct expanded form of -x(x - 7)?

-x^2 - 7x

x^2 - 7x

-x^2 + 7x (correct)

x^2 + 7x

Which expression represents the area of the picture centered in a frame with a length of (x + 30) cm and width of x cm?

$2x + 30$

$x^2 + 30x$ (correct)

$x^2 + 30$

$x^2 + 60x$

What is the error in the expression (x - 9)(x + 9) = x^2 - 18x - 81?

It should equal x^2 - 81. Signup and view all the answers

If a pool is 5 m longer than it is wide, what is its width if the dimensions of the pool are represented by x m?

x m Signup and view all the answers

What expression represents the entire area of the garden including the pond?

10 + x(x + 3) Signup and view all the answers

How can you express the area of the garden excluding the pond in simpler terms?

x^2 + 3x - 10 Signup and view all the answers

If the area of the garden excluding the pond is represented as x^2 + 3x - 10, which of these factors represents it when simplified?

(x - 5)(x + 2) Signup and view all the answers

What is the area of the garden excluding the pond when x = 5?

10 m^2 Signup and view all the answers

When x = 7, what is the value of the expression for the area of the garden excluding the pond?

24 m^2 Signup and view all the answers

What is the expanded form of $a(b + c)$ using the distributive law?

$ab + ac$ Signup and view all the answers

Which of the following expressions is considered a binomial product?

$(a + b)(c + d)$ Signup and view all the answers

What is the expanded form of $(a - b)^2$?

$a^2 - 2ab + b^2$ Signup and view all the answers

Which of the following represents the difference of perfect squares?

$(a + b)(a - b)$ Signup and view all the answers

What is the result of $x(x + 2)$ using the distributive law?

$x^2 + 2x$ Signup and view all the answers

Which expression is NOT a perfect square?

$4x^2 - 25$ Signup and view all the answers

What is the outcome of $(a + b)(c + d)$?

$ab + ac + bc + bd$ Signup and view all the answers

Which of the following correctly expresses the formula for a perfect square $(a + b)^2$?

$a^2 + 2ab + b^2$ Signup and view all the answers

What is the expression for the area of the pool given as $xm$?

$xm^2$ Signup and view all the answers

What does the total area include when the pool size is $xm$ and the paving is $(x + 5)m$?

The area of the pool plus the paving area Signup and view all the answers

Which of the following represents the expanded expression of a paved area if its dimensions are $xm$ for the pool and $(x + 5)m$ for the paved area?

$mx^2 + 5mx$ Signup and view all the answers

What is the area of the pool when $x = 4$ and the pool size is defined as $xm$?

$16m$ Signup and view all the answers

If the paving area is defined as $(x + 5)m$, what is the paved area when $x = 4$?

$45m$ Signup and view all the answers

Which method is used to write an expression like $x^2 - 9$ as a product of its factors?

Difference of perfect squares Signup and view all the answers

What is a common factor in the expression $2x + 6$ that allows for factorization?

2 Signup and view all the answers

Which of the following expressions can be factored as a difference of perfect squares?

$x^2 - 16$ Signup and view all the answers

What is the correct expression for the length of a rectangle if the width is represented by x?

(x + 3) m Signup and view all the answers

If the width of the rectangle is 4 m, what is the area of the rectangle?

28 m² Signup and view all the answers

What is the first step in finding the dimensions of the rectangle given its area and length in relation to the width?

Write an equation for area using the given dimensions. Signup and view all the answers

When solving for x in the equation $x(x + 3) = 28$, what is the resulting quadratic equation?

$x^2 + 3x - 28 = 0$ Signup and view all the answers

What is the value of x that must be chosen for the dimensions of the rectangle given $x(x + 3) = 28$?

4 Signup and view all the answers

Which equation correctly represents the area of a rectangle if its area is fixed at 72 m² and length is 6 m more than its width?

$x(x + 6) = 72$ Signup and view all the answers

In the problem where a triangle's area is $4 cm^2$, if the height is 2 cm more than the base, which expression correctly represents the area?

$A = rac{1}{2}b(b + 2)$ Signup and view all the answers

If a rectangle has an area of 24 m² and its length is 5 m longer than its width, what is the correct equation to find the width?

$x(x + 5) = 24$ Signup and view all the answers

Study Notes

Distributive Law

The distributive law expands brackets by multiplying each term inside the bracket by the term outside the bracket.
Example: a(b + c) = ab + ac
Example: (a + b)(c + d) = ac + ad + bc + bd
Example: a(b − c) = ab − ac

Perfect Squares

A perfect square is the result of squaring a binomial.
(a + b)² = (a + b)(a + b) = a² + ab + ba + b² = a² + 2ab + b²
(a − b)² = (a − b)(a − b) = a² − ab − ba + b² = a² − 2ab + b²

Difference of Perfect Squares (DOPS)

A DOPS is the result of multiplying two binomials that are the same except for the sign between the terms.
(a + b)(a − b) = a² − ab + ba − b² = a² − b²

Factoring a Difference of Perfect Squares

Factoring a DOPS is the reverse process of expanding a DOPS.
The difference of two perfect squares can be factored into the product of two binomials: one with a plus sign and one with a minus sign.
Example: x² - 9 = (x + 3)(x - 3)

Applications of Quadratics

Quadratic equations are often used to model real-world situations involving area, perimeter, and other geometric concepts.
By setting up an equation and solving for the unknown variable, we can find unknown dimensions or other quantities.
Problems involving areas of rectangles, areas of triangles, and areas of figures with squares and rectangles are common applications of quadratics.
Quadratic equations are used to determine the dimensions of a rectangle with a given area and a relationship between its length and width.
Be careful to disregard solutions that do not make sense in context, such as negative dimensions or zero width.

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Description

This quiz covers key concepts in Algebra Class 10, including the distributive law, perfect squares, and the difference of perfect squares (DOPS). It provides examples and applications of quadratic equations. Test your understanding of these fundamental algebraic principles.