For two distinct numbers \(p\) and \(q\), it is known that \(p^2 = 5p - 3\) and \(q^2 = 5q - 3\). Which of the following equations has roots \(\frac{p}{q}\) and \(\frac{q}{p}\)?

In an algebra practice set, two quadratic equations are given as \(x^2 + ax + b = 0\) and \(x^2 + bx + a = 0\). The difference between the two roots of the first equation is the same as the difference between the two roots of the second equation. If \(a \ne b\), which of the following must be true?

the numbers \(p\) and \(q\) are themselves the roots of the quadratic equation \(x^2 + px + q = 0\). Which of the following values of \(p\) and \(q\) satisfies this condition?

In a quadratic equation practice set, the equation is given as \((a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0\). For what value of \(a\) is one root of this equation twice the other root?

the sum of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is equal to the sum of the squares of the reciprocals of the roots. Then \(\frac{a}{c}\), \(\frac{b}{a}\), and \(\frac{c}{b}\) are in:

the equation \(x^2 - 3|x| + 2 = 0\) is given. How many real solutions does this equation have?

two positive numbers have arithmetic mean \(9\) and geometric mean \(4\). Which quadratic equation has these two numbers as its roots?

\((1 - p)\) is given as one root of the quadratic equation \(x^2 + px + (1 - p) = 0\). What are the roots of this equation?

one root of the equation \(x^2 + px + 12 = 0\) is \(4\). Also, the equation \(x^2 + px + q = 0\) has equal roots. What is the value of \(q\)?

the roots of the quadratic equation \(x^2 - bx + c = 0\) are two consecutive integers. What is the value of \(b^2 - 4c\)?

For a real value of \(k\), both roots of the quadratic equation \(x^2 - 2kx + k^2 + k - 5 = 0\) are less than \(5\). Then \(k\) lies in which of the following intervals?

both roots of the equation \(x^2 - 2mx + m^2 - 1 = 0\) are greater than \(-2\) but less than \(4\). All possible values of \(m\) lie in which of the following intervals?

the difference between the roots of the equation \(x^2 + ax + 1 = 0\) is less than \(\sqrt{5}\). What is the set of possible values of \(a\)?

Two quadratic equations in a CAT algebra exercise are \(x^2 - 6x + a = 0\) and \(x^2 - cx + 6 = 0\). They have one root in common. The remaining roots of the first and second equations are integers in the ratio \(4:3\). What is the common root?

he equation \(x^2 - 7|x| - 18 = 0\) is given. What is the number of real solutions of this equation?

In a CAT algebra drill, the quadratic equation \(x^2 - 9x + |k| = 0\) has real roots. How many integer values can \(k\) take?

the equation \(x^2 - 11x + |p| = 0\) has integer roots. How many integer values can \(p\) take?

the equation \(x^2 + 5x - 7 = 0\) has roots \(a\) and \(b\). Another equation \(2x^2 + px + q = 0\) has roots \(a + 1\) and \(b + 1\). Find the value of \(p + q\).

for a quadratic equation, the product of the roots is \(5\) more than the sum of the roots. If one root is \(1\) more than the other, what can be the product of the roots?

the equation \(x^2 - 7|x| - 30 = 0\) is given. How many real solutions does this equation have?

the quadratic expression \(x^2 + (2p + 1)x + p^2 - 1\) becomes \(0\) for some real value of \(x\). For which values of \(p\) is this possible?

the roots \(\alpha\) and \(\beta\) of the equation \(3x^2 + \lambda x - 1 = 0\) satisfy \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15\). Find the value of \((\alpha^3 + \beta^3)^2\).

Three quadratic equations \(x^2 + mx + 9 = 0\), \(x^2 + nx + 17 = 0\), and \(x^2 + (m+n)x + 35 = 0\) have a common negative root. Find the value of \(2m + 3n\).

Let \(m\) and \(n\) be positive integers. If the equations \(x^2 + mx + 2n = 0\) and \(x^2 + 2nx + m = 0\) have real roots, then what is the smallest possible value of \(m+n\)?

\(x\) and \(a\) are integers satisfying \(x^2 - 2|x| + |a - 2| = 0\). In how many ways can the ordered pair \((x,a)\) be chosen?

the equation \(\left(x+\frac{1}{x}\right)^2 - 3\left(x+\frac{1}{x}\right) + 2 = 0\) is given. What is the number of distinct real roots of this equation?

\(A\) is a real number. The roots of the equation \(x^2 - 4x - \log_2 A = 0\) are real and distinct if and only if:

the quadratic equation \(x^2 + bx + c = 0\) has roots \(4a\) and \(3a\), where \(a\) is an integer. Which of the following is a possible value of \(b^2 + c\)?

the equation \(|x|(6x^2 + 1) = 5x^2\) is given. How many real solutions does this equation have?

the equation \(|x^2 - x - 6| = x + 2\) is given. What is the product of its distinct real roots?

the equation \(x^2 + (a+3)x - (a+5) = 0\) has roots \(p\) and \(q\). What is the minimum possible value of \(p^2 + q^2\)?