Practice Questions

Define a quadratic equation and provide one example.

Solve the equation $2x^2 - x + \frac{1}{8} = 0$ by factorisation.

Recall the quadratic formula used to find the roots of the equation $ax^2 + bx + c = 0$.

Identify the values of $a, b,$ and $c$ for the quadratic equation $3x - 5x^2 + 9 = 0$ after writing it in standard form.

Without solving, examine the nature of the roots of the quadratic equation $3x^2 - 5x + 2 = 0$.

State the condition on the discriminant for a quadratic equation to have no real roots.

Recall the standard form of a quadratic equation.

Analyze the equation $(x+1)^3 = x^3 + 5x + 7$ to determine if it is a quadratic equation. Justify your answer.

Name the expression $b^2 - 4ac$ associated with the quadratic equation $ax^2 + bx + c = 0$.

The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was 124. Apply this information to calculate their present ages.

Examine if it is possible to design a rectangular garden whose length is 4 meters more than its width and whose area is $140 \text{ m}^2$. If possible, calculate its dimensions.

Explain the significance of the condition $a \neq 0$ in the definition of a quadratic equation $ax^2 + bx + c = 0$.

Describe the three possible types of real roots a quadratic equation can have.

Solve the quadratic equation $x^2 - (1+\sqrt{2})x + \sqrt{2} = 0$ by factorisation.

Solve the quadratic equation $4x^2 - 4ax + (a^2 - b^2) = 0$ for $x$.

Explain what is meant by a 'root' of a quadratic equation.

Describe the maximum number of roots a quadratic equation can have and why.

For the quadratic equation $3x^2 - 7x + 2 = 0$, list the values of $a, b, c$ and calculate its discriminant.

Analyze the equation $x(2x+3) = x^2+1$ to determine if it is a quadratic equation and find its roots if it is.

A motorboat, whose speed is $18 \text{ km/h}$ in still water, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Formulate a quadratic equation for the speed of the stream. Let the speed of the stream be $x \text{ km/h}$.

For the quadratic equation $16x^2 - 8x + 1 = 0$, evaluate which method of solving—factorisation or the quadratic formula—is more efficient. Justify your choice.

Is it possible to design a rectangular park with a perimeter of $100$ m and an area of $700 \text{ m}^2$? Justify your answer by formulating a quadratic equation and evaluating its discriminant, without solving for the dimensions.

A student states that the equation $x^2 + 9 = 0$ has no roots. Critique this statement.

Create a quadratic equation of the form $ax^2 + bx + c = 0$ where the coefficients $a, b, c$ are integers, and whose roots are $2 + \sqrt{3}$ and $2 - \sqrt{3}$.

Explain the relationship between the zeroes of a quadratic polynomial and the roots of a quadratic equation.

Explain the general procedure to determine if a given algebraic equation is quadratic or not.

Apply the concept of quadratic equations to find two consecutive odd positive integers, the sum of whose squares is 290.

Calculate the value(s) of $k$ for which the quadratic equation $(k+4)x^2 + (k+1)x + 1 = 0$ has equal roots.

A piece of wire of length 40 cm is bent into the form of a rectangle. Calculate the dimensions of the rectangle if its area is $96 \text{ cm}^2$.

Critique the following solution for finding the roots of $x^2 - 5x + 6 = 0$. Student's work: "The equation is $x^2 - 5x + 6 = 0$. I need two numbers that add to -5 and multiply to 6. These are -2 and -3. So the roots are $x = -2$ and $x = -3$." Identify the error in the reasoning and provide the correct solution.

Design a word problem involving the ages of two people that can be modeled by the quadratic equation $x^2 + 5x - 300 = 0$, where $x$ represents the current age of the younger person.

Identify if the equation $(x+1)^3 = x^3 + 2x + 1$ is a quadratic equation. Explain your reasoning.

Propose a quadratic equation whose roots are reciprocals of the roots of $2x^2 - 3x - 5 = 0$, without finding the roots of the original equation.

Evaluate the statement: "An equation of the form $(x+a)^3 - (x-a)^3 = 0$ is always a quadratic equation for any non-zero real number $a$." Justify your conclusion by simplifying the general form of the equation.

Consider the equation $x^2 + 2(k+1)x + (k^2+1) = 0$. Evaluate the nature of its roots based on different real values of the parameter $k$. Determine for which values of $k$ the equation has real roots, and for which values it has no real roots.

A company designs a rectangular billboard. The marketing team proposes that the length should be 3 meters more than its width. The design team mandates that a 1-meter wide border must be painted on all sides within the billboard area. If the area of this inner border is required to be exactly $22 \text{ m}^2$, formulate a quadratic equation to find the dimensions of the billboard. Do not solve the equation.

Compare the equations $(x-a)(x-b) = m^2$ and $(x-c)(x-d) = n^2$. Analyze the condition for the first equation to have real roots for all values of $m$.

A student claims that for the quadratic equation $(k-1)x^2 + 2kx + 4 = 0$, it is possible to find a value of $k$ such that the equation has two distinct real roots, two equal real roots, and no real roots. Justify this claim by finding the range of values for $k$ for each case.

Summarize how the discriminant, $D = b^2 - 4ac$, determines the nature of the roots for a quadratic equation $ax^2 + bx + c = 0$.

Justify why for a quadratic equation $ax^2+bx+c=0$ with rational coefficients, if one root is $p + \sqrt{q}$ (where $p, q$ are rational and $\sqrt{q}$ is irrational), then the other root must be its conjugate, $p - \sqrt{q}$.

Solve for $x$: $\frac{1}{x+4} - \frac{1}{x-7} = \frac{11}{30}$, given $x \neq -4, 7$.

Design a scenario for a business that involves calculating a break-even point (where profit is zero). The scenario should lead to the quadratic equation $P(x) = -5x^2 + 100x - 420 = 0$, where $x$ is the number of units sold (in thousands) and $P(x)$ is the profit (in thousands of rupees). Propose what the values in the equation could represent in the business context.

Justify the condition on the coefficients $a, b,$ and $c$ of the quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$) such that its roots are equal in magnitude but opposite in sign.

A motorboat whose speed is $18 \text{ km/h}$ in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Calculate the speed of the stream.

Demonstrate that the equation $(x^2+1)^2 - x^2 = 0$ has no real roots by analyzing its discriminant.