C-05 — Equation

The standard form of a quadratic equation in one variable is —

The values of variables for which both sides of an equation are equal are called —

In $ax^2 + bx + c = 0$, which condition must hold?

The two roots of $ax^2 + bx + c = 0$ are given by —

The expression $b^2 - 4ac$ in the quadratic formula is called —

For $ax^2 + bx + c = 0$ with $b = 0$, the values of $x$ are —

For $ax^2 + bx + c = 0$ with $c = 0$, the roots are —

The roots of $x^2 - 5x + 6 = 0$ are —

The discriminant of $x^2 - 6x + 9 = 0$ is —

The roots of $x^2 - 6x + 9 = 0$ are —

The roots of $x^2 - 2x - 2 = 0$ are —

The roots of $3 - 4x - x^2 = 0$ are —

If $b^2 - 4ac > 0$ and is a perfect square, the roots of $ax^2 + bx + c = 0$ are —

If $b^2 - 4ac > 0$ but not a perfect square, the roots are —

If $b^2 - 4ac = 0$, the roots are —

When $b^2 - 4ac = 0$, each root equals —

If $b^2 - 4ac < 0$, the two roots are —

Which condition gives real, unequal and irrational roots?

Consider the following statements about a quadratic $ax^2+bx+c=0$ with real coefficients:

Nature of the roots of $x^2 - x - 11 = 0$ is —

The roots of $2x^2 + 9x + 9 = 0$ are —

The discriminant of $4x - 1 - x^2 = 0$ is —

The roots of $4x - 1 - x^2 = 0$ are —

The roots of $2x^2 - 5x - 1 = 0$ are —

The roots of $3x^2 + 7x + 1 = 0$ are —

The roots of $x^2 - 8x + 16 = 0$ are —

The roots of $2x^2 + 7x - 1 = 0$ are —

The roots of $7x - 2 - 3x^2 = 0$ are —

The roots of $2 - 3x^2 + 9x = 0$ are —

In an equation involving square roots, freeing the radical sign by squaring may introduce —

Roots that arise from the squared equation but do not satisfy the original equation are called —

After solving a radical equation by squaring, we must —

The solution of $\sqrt{8x+9} - \sqrt{2x+15} = \sqrt{2x-6}$ is —

The solutions of $\sqrt{2x+8} - 2\sqrt{x+5} + 2 = 0$ are —

The acceptable solution of $\sqrt{2x+9} - \sqrt{x-4} = \sqrt{x+1}$ is —

In $\sqrt{2x+9} - \sqrt{x-4} = \sqrt{x+1}$, the value $x = -5$ is rejected because —

The solutions of $\sqrt{(x-1)(x-2)} + \sqrt{(x-3)(x-4)} = \sqrt{2}$ are —

To solve $\sqrt{x^2-6x+15} - \sqrt{x^2-6x+13} = \sqrt{10} - \sqrt{8}$, the most useful substitution is —

The solutions of $\sqrt{x^2-6x+15} - \sqrt{x^2-6x+13} = \sqrt{10} - \sqrt{8}$ are —

The solution of $(1+x)^{1/3} + (1-x)^{1/3} = 2^{1/3}$ is —

To begin solving $\sqrt{x^2 - 8} + \sqrt{x^2 - 14} = 6$, a useful substitution is —

To solve $\sqrt{x^2 - 6x + 9} - \sqrt{x^2 - 6x + 6} = 1$, replacing $x^2 - 6x + 6 = y$ gives —

Squaring an equation involving radicals usually requires —

An equation in which the unknown variable exists as an exponent is called —

Which of the following is an indicial equation?

The property used to solve $a^x = a^m$ requires —

If $a > 0,\ a \neq 1$ and $a^x = a^m$, then —

$4096$ expressed as a power of $2$ is —

$729$ expressed as a power of $3$ is —

The solution of $2^{x+7} = 4^{x+2}$ is —

The solution of $3 \cdot 27^x = 9^{x+4}$ is —

The solution of $3^{mx-1} = 3a^{mx-2}\ (a > 0,\ a \neq 3,\ m \neq 0)$ is —

The solution of $2^{2x-3} \cdot a^{x-2} = 2^{x-3} \cdot 4a^{1-x}\ (a > 0,\ a \neq 1/2)$ is —

The solution of $a^{-x}(a^x + b^{-x}) = 1 + \dfrac{1}{(ab)^2}\ (a>0,\ b>0,\ ab \neq 1)$ is —

The solution of $3^{x+5} = 3^{x+3} + \dfrac{8}{3}$ is —

The substitution used in solving $3^{2x-2} - 5 \cdot 3^{x-2} - 66 = 0$ is —

The solution of $3^{2x-2} - 5 \cdot 3^{x-2} - 66 = 0$ is —

After substituting $p = a^x$, the equation $a^{2x} - (a^2+1)a^x + a^2 = 0$ becomes —

The solutions of $a^{2x} - (a^2+1)a^x + a^2 = 0\ (a > 0,\ a \neq 1)$ are —

The solution of $5^x + 5^{2-x} = 26$ is —

The solution of $4^{1+x} + 4^{1-x} = 10$ is —

The solution of $3(9^x - 4 \cdot 3^{x-1}) + 1 = 0$ is —

To solve $5^x + 5^{2-x} = 26$, the most natural substitution is —

A pair $(x,y) = (a,b)$ is a solution of a system if —

The solutions of $x + \dfrac{1}{y} = \dfrac{5}{2},\ y + \dfrac{1}{x} = 3$ are —

From $x^2 = 3x + 6y$ and $xy = 5x + 4y$, factoring after subtraction gives —

One solution of $x^2 + y^2 = 61,\ xy = -30$ is —

One solution of $x^2 - 2xy + 8y^2 = 8,\ 3xy - 2y^2 = 4$ is —

From $\dfrac{x+y}{x-y} + \dfrac{x-y}{x+y} = \dfrac{5}{2}$ together with $x^2 + y^2 = 90$, we obtain —

The solutions of the system in the previous question are —

The solutions of $x^2 + y^2 = 25,\ xy = 12$ include —

The solutions of $x^2 = 3x + 2y,\ y^2 = 3y + 2x$ include —

The solutions of $x^2 + xy + y^2 = 3,\ x^2 - xy + y^2 = 7$ include —

The sum of areas of two square regions is $650$ sq m and the rectangle on their sides has area $323$ sq m. The sides are —

Twice the breadth of a rectangle is $10$ m more than its length and the area is $600$ sq m. The length is —

A two-digit number divided by the product of its digits gives quotient $3$, and adding $18$ reverses the digits. The number is —

Sum of areas of two squares is $481$ sq m and the rectangle on their sides has area $240$ sq m. The sides are —

The sum of the squares of two positive numbers is $250$ and their product is $117$. The numbers are —

The diagonal of a rectangle is $10$ m. The rectangle whose sides are the sum and difference of the original sides has area $28$ sq m. The original sides are —

The sum of squares of two numbers is $181$ and their product is $90$. The difference of their squares is —

A rectangle has area $24$ sq m. Another rectangle whose length and breadth are $4$ m and $1$ m more respectively has area $50$ sq m. The first rectangle's sides are —

Twice the breadth of a rectangle is $23$ m more than its length and the area is $600$ sq m. The length and breadth are —

The perimeter of a rectangle is $8$ m more than the sum of its diagonals. If the area is $48$ sq m, the length and breadth are —

A two-digit number divided by the product of its digits gives quotient $2$, and adding $27$ reverses the digits. The number is —

A rectangular garden has perimeter $56$ m and one diagonal $20$ m. The side of a square enclosing an equal area is —

A rectangular field has area $300$ sq m and its semi-perimeter is $10$ m more than a diagonal. The length and breadth are —