Quadratic Equations

If the quadratic equation $x^2 = 3x - 2$ is written in the standard form $ax^2 + bx + c = 0$, what is the value of the sum $a + b + c$?

The standard form of the equation $(x-3)^2 = 0$ is:

Represent the statement 'The sum of a number and its square is 20' as a quadratic equation.

What is the value of $a$ in the equation $x(x-1) = 0$?

If the equation $2(x - 3)^2 = 3(x + 1) - 4$ is written in the standard form $ax^2 + bx + c = 0$, what is the value of the expression $a - b + c$?

The sum of the squares of two consecutive multiples of 7 is 637. If the smaller multiple is $x$, represent this situation as a quadratic equation in the form $x^2 + bx + c = 0$.

Identify the standard form of the equation $(x+1)(x+2) = 3$.

An aeroplane left $50$ minutes later than its scheduled time, and in order to reach its destination $1250\text{ km}$ away in time, it had to increase its speed by $250\text{ km/h}$ from its usual speed. If $x$ represents the usual speed of the aeroplane in $\text{km/h}$, find the quadratic equation in standard form $ax^2 + bx + c = 0$ (with $a=1$) that models this situation.

If one root of $ax^2 + bx + c = 0$ is three times the other, then $3b^2$ is equal to:

Find the standard form of the equation $(\sqrt{x-1})^2 + x = 5$.

Determine the roots of $x^2 - 100 = 0$.

Which of the following are the roots of $x^2 + 12x + 27 = 0$?

Solve for $x$: $x^2 - 11x + 28 = 0$.

Find the roots of $x^2 - 17x + 72 = 0$.

Solve the equation $x^2 - 11x + 28 = 0$ by factorisation.

Solve for $x$: $4x^2 - 2(a^2+b^2)x + a^2b^2 = 0$.

Find the roots of the quadratic equation $21x^2 - 8x - 4 = 0$.

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

Factorise the quadratic equation $x^2 - (\sqrt{2} + \sqrt{3})x + \sqrt{6} = 0$ to find the roots.

Solve for $x$ by factorisation: $2x^2 + (2a + b)x + ab = 0$.

If $D=0$ for the equation $x^2 + kx + 9 = 0$, find the positive value of $k$.

Find the roots of $x^2 - 16x + 64 = 0$.

The discriminant of $x^2 + 11x + 24 = 0$ is:

Roots of $2x^2 - 3x - 2 = 0$ are:

Find the roots of $4x^2 - 4x - 3 = 0$ using the quadratic formula.

Solve $x^2 - 3x + 1 = 0$ for $x$ by applying the quadratic formula.

What are the solutions for $6x^2 - x - 2 = 0$ using the quadratic formula?

Solve for $x$: $3a^2x^2 + 8abx + 4b^2 = 0$ using the quadratic formula.

Determine the roots of $x^2 - 2px + p^2-q^2 = 0$ using the quadratic formula.

Solve the quadratic equation $x^2 - (2a-1)x + a^2-a-6 = 0$ for $x$.

For the equation $x^2 - 2x + k = 0$, find $k$ if the roots are equal.

If $D = -16$ for a quadratic equation, the nature of roots is:

The roots of $3x^2 + 7x + 2 = 0$ are:

For $x^2 + 2kx + k = 0$ to have equal roots, the non-zero value of $k$ is:

Find the values of $k$ for which the equation $x^2 + 2(k+2)x + 9k = 0$ has equal roots.

The nature of the roots for the quadratic equation $5x^2 - 2x - 3 = 0$ is:

What is the discriminant of the quadratic equation $x^2 + x + 0.25 = 0$?

If $a, b, c$ are distinct positive real numbers, the roots of $ax^2 + 2bx + c = 0$ and $bx^2 + 2cx + a = 0$ cannot both be:

For what value of $k$ is the discriminant of $x^2 + kx + (k + 3) = 0$ equal to $0$?

If $x^2 + px + 1 = 0$ has non-real roots and $x^2 + qx + 1 = 0$ has real roots, then which is true for $p$ and $q$?

The product of two consecutive positive integers is $240$. Find the larger integer.

A takes $6$ days less than the time taken by B to finish a piece of work. If both A and B together can finish it in $4$ days, find the time taken by B to finish the work alone.

The perimeter of a right triangle is $60\text{ cm}$ and its hypotenuse is $25\text{ cm}$. Find the area of the triangle.

A girl is twice as old as her sister. Four years hence, the product of their ages (in years) will be $160$. Find their present ages.

A two-digit number is such that the product of its digits is $12$. When $36$ is added to the number, the digits interchange their places. What is the original number?

A square piece of tin of side $x$ cm is used to make an open box by cutting a square of $5$ cm from each corner and folding up the flaps. If the volume of the resulting box is $500$ cm$^3$, what is the length of the side of the original square piece of tin?

A cyclist travels $24$ km at a certain speed. If he increases his speed by $2$ km/h, he takes $2$ hours less to cover the distance. Find his original speed.

The total surface area of a closed rectangular box with a square base is $264\text{ cm}^2$. If the height of the box is $2\text{ cm}$ more than the length of a side of the square base, what is the volume of the box?

A farmer wants to enclose a rectangular vegetable garden with an area of $1250\text{ m}^2$. He uses a straight stone wall for one side of the garden and $100\text{ m}$ of fencing for the remaining three sides. Find the length of the side of the garden that is parallel to the stone wall.

In a local chess tournament, every participant plays exactly two matches against every other participant. If the total number of matches played in the tournament is $110$, find the number of participants.

The discriminant of $x^2 + 5x + 5 = 0$ is:

What is the value of $c$ in the equation $4x^2 - 1 = 0$?

Is $x^2 + 1/x^2 = 2$ a quadratic equation?

The product of the roots of $x^2 - 4 = 0$ is:

Find the roots of the equation $x + \frac{1}{x} = 2.5$.

What constant must be added to $x^2 - 8x$ to make it a perfect square trinomial?

Convert the equation $(2x + 1)(3x + 2) = 6(x - 1)(x - 2)$ into the standard form $ax^2 + bx + c = 0$.

A rectangular storage area of $200$ $m^2$ is to be enclosed by a fence and then divided into two equal parts by an internal fence parallel to one of its sides. If the total length of the fencing used (including the internal fence) is $70$ m and $x$ represents the length of the internal fence, which quadratic equation represents this situation?

If the equation $px^2 + qx + r = 0$ represents a quadratic equation, which of the following must be true?

Represent the following as a quadratic equation: The product of two consecutive positive integers is $306$.

Determine the roots of $x^2 + 5x - 6 = 0$.

Solve for $x$: $x^2 - 18x + 81 = 0$.

Find the roots of $x^2 - 81 = 0$.

What are the roots of $x^2 + 9x + 20 = 0$?

Find the roots of the quadratic equation $4x^2 + 4\sqrt{3}x + 3 = 0$ using factorisation.

Determine the roots of the equation $21x^2 - 8x - 4 = 0$ by splitting the middle term.

Solve the quadratic equation $x^2 - 4ax + (4a^2 - 9b^2) = 0$ for $x$ using the factorisation method.

Solve the equation for $x$: $x^2 - (a+b)x + ab = 0$.

Find the roots of the equation $\frac{1}{x-3} + \frac{2}{x-2} = \frac{8}{x}$.

Solve for $x$: $x^2/a^2 - 3x/a + 2 = 0$.

If $b^2 - 4ac > 0$, the quadratic equation has:

What is the nature of the roots of $3x^2 + x - 1 = 0$?

Find the discriminant of $4x^2 - 4x + 1 = 0$.

The nature of the roots of $x^2 - 12x + 36 = 0$ is:

Find the value of $k$ for which the quadratic equation $(k^2 + 1)x^2 - 2(k + 3)x + 1 = 0$ has real and equal roots.

If the equation $x^2 - kx + 4 = 0$ has real roots, the value of $k$ can be:

The quadratic equation $x^2 - 2x + 1 = 0$ has roots that are:

If $a, b, c$ are in AP, then the roots of $ax^2 + bx + c = 0$ are real if:

For the equation $x^2 + 2(k-4)x + 2k = 0$ to have equal roots, the values of $k$ are:

If the roots of $ax^2 + bx + c = 0$ are real and of opposite signs, then: