Quadratic equations practice · Mixed difficulty

Option B (${x^2} < 2$) is correct because substituting y = a/b into the equation for x and using the given condition ${y^2} > 2$ leads to ${x^2} < 2$.

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

Option D ($\frac{9}{4}\left(9-p^{2}\right)$) is correct because it is derived by using the relationships between the roots and coefficients of the given quadratic equations and simplifying the expression.

Option B (8) is correct because it is the only solution that satisfies the equation $2x - 1 = |x + 7|$.

The correct answer is A ($\leq 9$) because it is derived using the Cauchy-Schwarz inequality with the given condition $a+b+c=6$.

Option B (${x^2} - 2x + 16 = 0$) is correct because it represents the quadratic equation whose roots are p and q, derived from the given conditions $p + q = 2$ and ${p^4} + {q^4} = 272$.

Option A (0) is correct because if the roots are equal in magnitude but opposite in sign, their sum is zero, which relates directly to the coefficients of the quadratic equation.

Option A $\left(-16, \frac{{65}}{{4}}\right)$ is correct because the inequality $16-x^2 > |x-a|$ is satisfied by at least one negative value of $x$ when $a$ lies in this interval.

Option A ($\left( { - 1,\frac{{ - 1}}{3}} \right)$) is correct because it represents the interval obtained by solving the inequality $\left| {3x + 2} \right| < 1$.

Option C $-1$ is correct because it is the value of $\lambda$ calculated using the vertex form of the quadratic equation and the given highest point.

Option D (3) is correct because the expression simplifies to 3 using the identity $a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$ when $a+b+c=0$, then $a^3 + b^3 + c^3 = 3abc$.

Option A (0) is correct because the equation has no solution as the LHS is always less than or equal to 2, while the RHS is always greater than or equal to 2, and they equal 2 at different x values.

Option C ($3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\ldots$) is correct because it satisfies the given conditions for the sum of the infinite geometric series and the sum of the cubes of the series.

Option C (Real with sum zero) is correct because the roots are 2 and -2, which are real and sum to zero.

Option C ($> (a+b+c)$) is correct because it follows from applying AM-GM inequality to the given expression.

Option A (1/4) is correct because it results from substituting the sum and product of roots into the simplified expression.

Option B (real and unequal) is correct because the discriminant of the quadratic equation is positive, indicating real and distinct roots.

Option C ($\frac{{{b^2} + 4ac}}{{{a^2}}}$) is correct because the difference between the roots remains constant even when the roots change by the same quantity.

Option A (4) is correct because solving for k and checking the discriminant condition gives k=4.

Option B ($9+\sqrt{3}$) is correct because it is the sum of the roots of the given equation after considering both cases of the absolute value.

Option C ($a, b$) is correct because $a$ and $b$ are the roots of the equation $(x-\alpha)(x-\beta)+c=0$ as derived from the given equation $(x-a)(x-b)=c$.

Option C ($\pm \;6$) is correct because it is the value of p that satisfies the condition where the difference between the roots of the given quadratic equation is 2.

The speeds of the car and bus are 50 km/hr and 25 km/hr respectively, which is obtained by setting up an equation based on the given time difference and solving for the speeds. Therefore, the correct answer is B ($25 {~km} / {hr}, 50 {~km} / {hr}$).

The correct answer is 7 because we need to find the number of pairs (p, q) such that $q^2 \geq 4p$ where $p, q \in \{1, 2, 3, 4\}$.

Option A (Imaginary) is correct because the discriminant of the quadratic equation is negative, implying imaginary roots.

Option C (20) is correct because after finding the equation of the plane and comparing it with the given form, the value of (a-b+c) is calculated to be 22.

Option B ($\alpha = 1\,\mathrm{and}\,\beta = - 2$) is correct because by substituting the roots into the equation and solving the resulting system of equations, we find these values for alpha and beta.

Option B (a+b=1, 0 < b < 1) is correct because using the tangent addition formula and the given conditions, we can derive the relationship a + b = 1 and the constraint on b.

The correct answer is B ($[9, \infty)$) because it represents the range of 'a' for which at least one root of the quadratic equation is greater than 2.

Option B ($d-c$) is correct because substituting x = c into the transformed equation yields a root of 0, implying that d-c is also a root.