Quadratic equations practice · Easy

Option C ($\{3,7\}$) is correct because these are the only integer values of 'a' that result in integer roots for the given equation.

Option D (-3) is correct because given the roots and the equation, the expression simplifies to -3 using the relationship between roots and coefficients.

The correct answer is 719 because it is the minimum value of a+b+c that satisfies the given conditions for the cubic polynomial f(x).

Option C (1) is correct because a=2 is the only value that simultaneously satisfies all three equations derived from the coefficients of the given identity.

Option D ($\frac{8}{5}$) is correct because by using the given condition $f(a)+f(b)=0$ and the fact that -1 is a root, we can find the other root of the quadratic expression.

Option A ($p = - 4,q = 7$) is correct because the sum and product of the roots of the quadratic equation are used to find the values of p and q, considering the complex conjugate root.

Option A (45) is correct because by finding the other root and using the relationship between roots and coefficients, we can determine the values of a and b and then calculate ab.

Option B ($\{4\}$) is correct because only 4 satisfies the given equation and the domain of the logarithmic function.

Option D ($\left\{ {125,8} \right\}$) is correct because both 125 and 8 are solutions to the given equation after substituting $y = x^{1/3}$ and solving the resulting quadratic equation.

The correct answer is A (45) because by finding the conjugate root and using Vieta's formulas, we can determine the values of a and b, and their product is 45.

Option B (1) is correct because the value of 'a' is found by using the ratio of roots and solving the resulting equation.

The correct answer is D ($\frac{{25}}{{81}}$) because it is the product of the roots of the given equation, found by solving for |x| and then finding the roots.

Option A (0) is correct because the exponential function ${e^{\left| x \right|}}$ is always greater than $\left| x \right|$, so they never intersect.

Option B ($x^2+4 a x+2 b=0$) is correct because it is derived by substituting the sum and product of the roots into the given expression and simplifying.

Option A ($\left[-1, \frac{{-1+\sqrt{5}}}{{2}}\right)$) is correct because by considering the domain and solving the inequality, we find the solution set to be $\left[-1, \frac{{-1+\sqrt{5}}}{{2}}\right)$.

Option C (${\left( {{a^3}c} \right)^{1/4}} + {\left( {a{c^3}} \right)^{1/4}} + b = 0$) is correct because it is derived from the given condition $\beta = {\alpha ^{1/3}}$ and the relationships between the roots and coefficients of the quadratic equation.

Option A ($a=b=c$) is correct because if all three quadratic equations have only positive roots, it implies that the coefficients must be equal.

Option B ($p=1$ or 0) is correct because substituting the roots into the given equation and solving the resulting system of equations yields these two possible values for p.

Option A (254) is correct because substituting the given values and simplifying the expression results in 254.

Option C (1, -4) is correct because by finding the values of 'b' and 'c' from the given roots and substituting them into the equation x² + bx + c = 0, we can solve for the roots.

Option D ($\frac{{ - 71}}{{256}}$) is correct because by using the centroid coordinates and the arithmetic progression property, we can find the values of a, b, and c, and then calculate the required expression using the roots of the quadratic equation.

Option C (${p^2} = q(q + 2)$) is correct because by using the sum and product of roots, and the trigonometric identity $\sin^2 \alpha + \cos^2 \alpha = 1$, we can derive the given relation.

Option C (5) is correct because using the given conditions $2b = a+c$ and $a^2 + b^2 + c^2 = 84$, and the triangle inequality, we can deduce that b = 5.

Option B ($1 + \sqrt 2$) is correct because it is the positive solution to the quadratic equation formed by the infinite continued fraction.

Option A ($k = \pm 4$) is correct because these values satisfy the condition for the two quadratic equations to have a common root, found by solving the system of equations.

Option A (0, 24) is correct because these are the values of 'a' that result in a common root for the two quadratic expressions.

Option D (None of these) is correct because the value of the expression is 4, which is not among the options.

Option A ($1 \le x \le 3$) is correct because by completing the square and analyzing the resulting equation, we can determine the range of possible values for x.

Option B ($\frac{5}{2}$) is correct because it is the product of the real roots obtained by solving the absolute value equation.

Option D ($- 3$) is correct because it is the only real root of the given cubic equation, found by factoring.