Quadratic equations practice · Mixed difficulty

Option B ((-5,-7)) is correct because the sum of the roots is -5 and -7 respectively, and the two common roots sum to zero.

Option B ($4\;ac > {b^2}$) is correct because it ensures the quadratic expression always has the same sign as c by guaranteeing no real roots.

Option B (3/2) is the correct answer because it satisfies the given exponential equation after simplification and taking logarithms.

Option B (5) is correct because the product of the roots of a quadratic equation is c/a, and if one root is the reciprocal of the other, their product is 1.

Option B ($9 x^2+2 x+1=0$) is correct because it is obtained by finding the sum and product of the new roots ($\frac{1}{\alpha^2}$ and $\frac{1}{\beta^2}$) and constructing the quadratic equation.

Option A (4) is correct because the equation has four solutions: $x = \pm 2$ and $x = \pm 3$.

Option A ($\sin ^{-1}\left(\tan \frac{\alpha}{2} \cdot|\cot \alpha|\right)$) is correct because it is derived by finding the angle between the vector product of two unit vectors and the plane containing the other two unit vectors.

Option B ([3/4, 1]) is correct because the expression \sin² θ + \cos⁴ θ lies within this interval, as determined by its maximum and minimum values.

Option A (AP) is correct because the condition for equal roots leads to the relationship $a + c = 2b$, which defines an arithmetic progression.

Option D (2) is correct because by substituting the given value of z and using the properties of logarithms and exponentials, we can find that cos z = 2.

Option A (1) is correct because the minimum value of the sum of the distances from z to 0 and z to 1 is the distance between 0 and 1, which is 1.

Option B (47) is correct because using the relationship between roots and coefficients and simplifying the expression yields the value 47.

Option D ($- \frac{1}{9},\; - 2$) is correct because these are the only solutions to the quadratic equation that also satisfy the domain of the function $y = \sqrt {x(x - 3)}$.

Option C ($\sqrt{3}$) is correct because it is the absolute value of b, derived from solving the equations with a common root.

Option B (${x^2} - 18x + 16 = 0$) is correct because the sum of the roots is 18 (twice the arithmetic mean) and the product of the roots is 16 (the square of the geometric mean).

Option D (18) is correct because solving the given conditions for p(x) and evaluating p(2) yields 18.

Option B (-1) is correct because by equating the coincident root in both equations, we can derive that a+b = -1.

Option B (At least one equation always has real roots) is correct because the sum of the discriminants is non-negative, implying at least one discriminant is non-negative.

Option D ($x \in \left[ { - 2,\; - 1} \right) \cup \left[ {3,\;4} \right)$) is correct because it represents the intervals where the greatest integer function satisfies the given equation.

Option A (Real) is correct because the discriminant of the quadratic equation is non-negative, implying that the roots are real.

Option B (0) is correct because the condition for common roots simplifies to $2r - p = 0$.

The correct answer is D $\left(1+\frac{{\sqrt{{5}}}}{{2}}, \infty\right)$ because it satisfies the condition that the quadratic expression is always greater than or equal to zero for all real x.

Option A ($a^2-b^2+2 a c=0$) is correct because it is the result of applying Vieta's formulas to relate the roots (sin α, cos α) to the coefficients of the quadratic equation and then using the trigonometric identity sin²α + cos²α = 1.

Option B $\ge 2$ is correct because it is derived using the AM-GM inequality.

Option B ($\left(p^3+q\right) x^2-\left(p^3-2 q\right) x+\left(p^3+q\right)=0$) is correct because it is derived by finding the sum and product of the roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ in terms of $p$ and $q$, and then constructing the quadratic equation.

Option C (0) is correct because the equation has no solution since the only possible solution, x=1, is an extraneous solution.

Option B (50) is correct because the integral simplifies to 50 after applying integration by parts and simplification.

Option C ($\frac{{q' - q}}{{p - p'}}$) is correct because it represents the common root derived from subtracting the two quadratic equations.

Option A ($7 / 8$) is correct because it is the minimum value of the quadratic equation calculated using the formula $\frac{{4AC-B^2}}{{4A}}$.

Option B ($b \ge a \ge c$) is correct because it ensures the discriminant of the quadratic equation is non-negative, guaranteeing real values for x.