What are the solutions to the quadratic equation x^2 - 7x + 12 = 0?
Quadratics And Complex Numbers
Question · Easy
Factor it
Short
Simplify the complex expression: (3 + 4i) + (5 - 2i).
What is the vertex of the parabola given by the equation y = (x - 3)^2 + 5?
Simplify i^14, where i is the imaginary unit.
What is the y-intercept of the quadratic function f(x) = 2x^2 - 4x + 6?
Would you rather: marbles
What is the value of the discriminant for the equation x^2 + 4x + 5 = 0, and what does it indicate about the roots?
Multiply the complex conjugate pair: (2 + 3i)(2 - 3i).
What is the axis of symmetry for the quadratic graph y = x^2 - 6x + 8?
Solve the quadratic equation 2x^2 - 5x - 3 = 0.
Question · Medium
Solve x^2 - 8x + 11 = 0 by completing the square.
Perform the multiplication: (4 - 3i)(1 + 2i).
Express the quotient (5 + i) / (1 - i) in standard a + bi form.
What are the complex roots of the equation x^2 + 6x + 13 = 0?
The height h (in feet) of a projectile launched upwards is modeled by h(t) = -16t^2 + 64t + 10, where t is time in seconds. What is the maximum height reached by the projectile?
Which list of transformations maps the parent graph f(x) = x^2 onto g(x) = -3(x + 2)^2 - 4?
Find the solution points to the system of equations: y = x^2 - 2x and y = 2x - 3.
Simplify the radical expression: sqrt(-48).
Write the vertex form of a parabola that has vertex (2, -1) and passes through the point (4, 7).
Find the absolute value (modulus) of the complex number z = 6 - 8i.
What are the zeroes of the function f(x) = 3x^2 - 12x + 12?
Solve x^2 + 4x + 20 = 0.
Rewrite y = 2x^2 + 12x + 11 in vertex form.
Evaluate the expression: (2i)^4 + 3i^9.
Solve the quadratic inequality: x^2 - 3x - 10 <= 0.
Question · Hard
For what values of k does the quadratic equation 4x^2 + kx + 9 = 0 have exactly one real solution?
Find all real integer solutions to the equation: (x^2 - 5)^2 - 2(x^2 - 5) - 8 = 0.
Simplify the quotient of complex numbers: (7 - 4i) / (3 + 2i).
A rectangular garden has an area of 54 square meters. The length is (2x + 1) meters and the width is (x + 2) meters. What is the value of x?
Evaluate the complex expression: (1 + i)^4.
What is the sum of the roots of the quadratic equation x^2 - 9x + 20 = 0?
What is the product of the roots of the quadratic equation x^2 + 5x - 14 = 0?
Simplify the expression with rational exponents: (8x^6)^(2/3).
Solve the radical equation: sqrt(3x + 1) = 5.
Solve sqrt(x + 6) = x and identify any extraneous solutions.
Simplify fully: sqrt(75 * x^5 * y^8), assuming x >= 0.
In a right-angled triangle, how is the sine ratio of an acute angle defined?
Similarity And Trigonometry
A right triangle has leg lengths of 9 cm and 12 cm. What is the length of the hypotenuse?
Triangle ABC is similar to triangle DEF. If side AB = 6 and corresponding side DE = 18, what is the scale factor from triangle ABC to triangle DEF?
In an isosceles right triangle (45-45-90), each leg measures 7 inches. What is the exact length of the hypotenuse?
In a 30-60-90 right triangle, the shorter leg opposite the 30-degree angle measures 5 cm. What is the exact length of the longer leg?
If sin(theta) = 3/5 for an acute angle theta in a right triangle, what is cos(theta)?
If two figures are similar with a scale factor of k = 4, by what factor does the area of the figure increase?
A 6-foot-tall student casts a shadow on level ground when the sun's angle of elevation is 30 degrees. What is the length of the shadow?
In right triangle ABC, an altitude is drawn to hypotenuse AB, dividing AB into segments of length 4 cm and 9 cm. What is the length of the altitude?
In right triangle ABC with right angle at C, side AC = 12 and side BC = 5. What is the measure of angle A to the nearest tenth of a degree?
From the top of a 150-foot lighthouse, the angle of depression to a boat out at sea is 25 degrees. To the nearest foot, how far is the boat from the base of the lighthouse?
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