Algebra Class: Word Problems and Equations

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Questions and Answers

What are two numbers whose sum is 27 and product is 182?

The two numbers are 9 and 18.

Find two consecutive positive integers whose sum of squares is 613.

The integers are 24 and 25.

In a right triangle where the altitude is 7 cm less than its base and the hypotenuse is 13 cm, what are the base and altitude?

The altitude is 5 cm and the base is 12 cm.

If a cottage industry produces pottery articles at a cost of Rs 3 more than twice the number produced, and the total cost is Rs 90, how many articles are produced?

20 articles are produced. Signup and view all the answers

Determine the dimensions of a rectangle with a perimeter of 28 meters and area of 40 square meters.

The dimensions are 8 meters by 6 meters. Signup and view all the answers

Find the base and altitude of a triangle where the base is 4 cm longer than its altitude and the area is 48 sq.cm.

The base is 12 cm and the altitude is 8 cm. Signup and view all the answers

If two trains are 50 km apart after 2 hours and one travels 5 km/hr faster than the other, what are the average speeds?

The first train's speed is 30 km/hr and the second's is 25 km/hr. Signup and view all the answers

In a class of 60 students where each boy contributed as many rupees as there are girls, and each girl contributed as many as boys to total Rs 1600, how many boys are there?

There are 40 boys in the class. Signup and view all the answers

What is the quadratic formula used for solving equations of the form ax² + bx + c = 0?

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Signup and view all the answers

What condition must be met for a quadratic equation to have real roots?

The condition is that $b^2 - 4ac > 0$. Signup and view all the answers

If the discriminant $b^2 - 4ac$ is less than zero, what does it indicate about the roots of the equation?

It indicates that the equation has no real roots. Signup and view all the answers

In the equation $2x^2 + x - 528 = 0$, what values correspond to a, b, and c?

Here, $a = 2$, $b = 1$, and $c = -528$. Signup and view all the answers

How would you express the area of a rectangular plot if the length is given as (2x + 1) m and the breadth as x m?

The area can be expressed as $x(2x + 1) = 528$ m². Signup and view all the answers

What does the expression $rac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ represent in the context of solving quadratic equations?

It represents the solutions or roots of the quadratic equation. Signup and view all the answers

What is the result of applying the quadratic formula if $b^2 - 4ac = 0$?

The equation will have exactly one real root. Signup and view all the answers

What expression is derived to find the roots of the quadratic equation by isolating x?

The roots can be expressed as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Signup and view all the answers

What are the roots of the equation $x^2 - 3x + 1 = 0$?

$\frac{3 + \sqrt{5}}{2}, \frac{3 - \sqrt{5}}{2}$ Signup and view all the answers

Determine the discriminant for the equation $x^2 - 3x + 1 = 0$.

$5$ Signup and view all the answers

What does the expression $b^2 - 4ac$ signify in the context of quadratic equations?

It is called the discriminant and indicates the nature of the roots of the quadratic equation. Signup and view all the answers

For the equation $(x - 2) - x = 3x(x - 2)$, what must be done to eliminate fractions?

Multiply through by $x(x - 2)$. Signup and view all the answers

In the context of the motor boat problem, what are the expressions for the speeds going upstream and downstream?

Upstream: $(18 - x)$ km/h, Downstream: $(18 + x)$ km/h. Signup and view all the answers

What role does the equation $24/(18 - x) - 24/(18 + x) = 1$ play in solving the motor boat problem?

It relates the time differences of the upstream and downstream journeys. Signup and view all the answers

If the speed of the stream is $x$ km/h, how does it affect the time taken for the motorboat to travel upstream?

It increases the time taken since the effective speed is decreased by $x$. Signup and view all the answers

What is the significance of the discriminant in a quadratic equation?

The discriminant, given by $b^2 - 4ac$, indicates the nature of the roots: two distinct real roots if greater than 0, one equal real root if equal to 0, and no real roots if less than 0. Signup and view all the answers

If the discriminant is negative, what can be concluded about the roots of the quadratic equation?

If the discriminant is negative, it indicates that the quadratic equation has no real roots and the roots are imaginary. Signup and view all the answers

How do the roots behave when the discriminant equals zero?

When the discriminant equals zero, the quadratic equation has two equal real roots, and the graph touches the x-axis at one point. Signup and view all the answers

Describe the shape of the quadratic graph when the discriminant is positive.

When the discriminant is positive, the graph of the quadratic equation forms a parabola that intersects the x-axis at two distinct points. Signup and view all the answers

Find the discriminant of the equation $x^2 - 6x + 9 = 0$ and state the nature of its roots.

The discriminant is $0$ (since $(-6)^2 - 4(1)(9) = 0$), thus the equation has two equal real roots. Signup and view all the answers

How do you identify whether a quadratic equation has imaginary roots?

A quadratic equation has imaginary roots if its discriminant is less than zero. Signup and view all the answers

In a quadratic equation $ax^2 + bx + c = 0$, what does it mean if $b^2 - 4ac = 0$?

It means that the quadratic equation has exactly one unique solution or two equal real roots. Signup and view all the answers

Given the quadratic equation $2x^2 - 4x + 3 = 0$, calculate the discriminant and discuss the nature of its roots.

The discriminant is $-8$ (since $(-4)^2 - 4(2)(3) = -8$), indicating that the roots are imaginary. Signup and view all the answers

What does the positive discriminant indicate about the roots of the quadratic equation x² + 7x - 60 = 0?

A positive discriminant indicates that the quadratic equation has two distinct real roots. Signup and view all the answers

Why can the solution x = -12 be disregarded in the context of the pole's distance from gate B?

The solution x = -12 can be discarded because distance cannot be negative. Signup and view all the answers

When applying the Pythagorean theorem, what does AB = 13 m imply about triangle APB?

AB = 13 m implies that AB is the hypotenuse of triangle APB. Signup and view all the answers

What is the significance of the equation AP - BP = 7 m in relation to the location of the pole?

The equation AP - BP = 7 m indicates that the pole is 7 m closer to gate B than to gate A. Signup and view all the answers

How does knowing the discriminant help when solving quadratic equations?

Knowing the discriminant helps predict the nature of the roots before attempting to solve it. Signup and view all the answers

Write a quadratic equation that has no real solution and provide its discriminant.

An example is x² + 4x + 5 = 0, with a discriminant of -4. Signup and view all the answers

What would be the discriminant value for a quadratic equation with exactly one real solution?

The discriminant value would be 0. Signup and view all the answers

State the nature of the roots for the equation 3x² - 2x + 1/3 = 0 when its discriminant is calculated.

The nature of the roots is complex since the discriminant is negative. Signup and view all the answers

If Vinay can complete painting in `x` days and Praveen in `x + 5` days, what is the equation to find `x`?

<code>1/x + 1/(x + 5) = 1/6</code> Signup and view all the answers

What is the formula for the sum of the roots of the quadratic equation `ax^2 + bx + c = 0`?

<code>-b/a</code> Signup and view all the answers

In the quadratic equation `ax^2 + bx + c = 0`, what expression represents the product of the roots?

<code>c/a</code> Signup and view all the answers

If the sum of a fraction and its reciprocal equals 2, what is the general representation of that fraction?

<code>x</code> where <code>x + 1/x = 2</code> Signup and view all the answers

What does it mean for a number `a` to be a root of a quadratic equation `ax^2 + bx + c = 0`?

<code>a^2 + ba + c = 0</code> Signup and view all the answers

What is the standard form of a quadratic equation?

<code>ax^2 + bx + c = 0</code> where <code>a ≠ 0</code> Signup and view all the answers

What method can also be used to solve a quadratic equation besides factorization?

<code>Completing the square</code> Signup and view all the answers

If `a > 0` or `a < 0`, how do these conditions affect the nature of roots in a quadratic equation?

<code>They determine whether roots are real or complex.</code> Signup and view all the answers

Study Notes

Quadratic Equations

A quadratic equation is an equation written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
The solutions (roots) to a quadratic equation can be found using various methods, including factoring, completing the square, and the quadratic formula.
The discriminant (b² - 4ac) of a quadratic equation helps determine the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (two equal roots).
- If b² - 4ac < 0, the equation has no real roots (two imaginary roots).
The quadratic formula is a general solution for finding the roots of any quadratic equation: x = (-b ± √(b² - 4ac)) / 2a

Quadratic Formula

The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0.
The formula provides a direct way to find the roots of the equation without needing to factor it.
The formula is x = (-b ± √(b² - 4ac)) / 2a
Where 'a', 'b', and 'c' are the coefficients in the quadratic equation.

Completing the Square

Completing the square is a method for solving quadratic equations.
The method involves rewriting the equation into a perfect square trinomial, thus allowing the use of square roots.
This method helps to isolate the 'x' variable and solve for it.

Factoring

Factoring is a method used to find the roots (solutions) to a quadratic equation.
It involves finding two expressions that, when multiplied together, result in the quadratic expression on the left side of the equation.
This method depends heavily on spotting the correct factors of the coefficients in the equation.

Nature of Roots

The discriminant is a part of the quadratic formula that can help predict the number of real roots an equation will have.
If the discriminant is positive, that means the quadratic equation has two distinct real roots.
If the discriminant is zero, that means the quadratic equation has two equal (or one distinct root).
If the discriminant is negative, that means the quadratic equation has two distinct complex (or imaginary roots).

Applications

Quadratic equations have extensive applications in real-world problems, often involving area, speed, and time calculations.
They are used to model projectile motion, optimize areas, or calculate the dimensions of various shapes.

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Description

This quiz covers various word problems related to algebra, including finding numbers based on their sum and product, solving for consecutive integers, determining dimensions of geometric shapes, and understanding relationships in a class. Test your problem-solving skills and knowledge of algebraic concepts with these diverse scenarios.