For what values of p does the quadratic equation x² + px + 1 = 0 have real roots?

The roots of the quadratic equation x² + ax + b = 0 are 1 and -2. Find the roots of the equation x² + (a+b)x + b = 0.

The roots of the equation x² - (k+1)x + (k+4) = 0 are real and equal. Find the value of k.

A piece of cloth costs Rs 30. If the piece were 1 m longer and each metre costs Re 1 less, the cost would remain unchanged. What is the length of the piece?

If the equation (m-1)x² - 2(m-1)x + 1 = 0 has equal roots, then find m.

A man buys a number of pens for Rs. 60. If he had bought 3 more pens for the same amount, each pen would have cost him Re. 1 less. How many pens did he buy?

If alpha and beta are the roots of the equation x² - 5x + 6 = 0, then the equation whose roots are 1/alpha and 1/beta is:

A farmer has 40 m of fencing material. He wants to enclose a rectangular plot of land using an existing wall as one side of the plot. If the area of the plot is 200 m², what are the dimensions of the plot?

If alpha and beta are the roots of the equation x² - 4x + 3 = 0, then the equation whose roots are 1/(alpha+1) and 1/(beta+1) is:

Total 50 questions have been generated with checks and revisions.Which of the following equations is NOT a quadratic equation?

The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

For what value of m does the quadratic equation (m-1)x² + 2(m-1)x + 1 = 0 have exactly one real root?

Revised Question 58: If alpha and beta are the roots of the equation x² - 7x + 10 = 0, find the value of alpha³ + beta³.

If alpha and beta are the roots of the equation x² - 7x + 10 = 0, find the value of alpha³ + beta³.

The product of two consecutive positive integers is 306. Find the integers.

Revised Question 56: If a is a root of the equation x² - x - 1 = 0, then the value of a³ - 2a + 1 is:

If a is a root of the equation x² - x - 1 = 0, then the value of a³ - 2a + 1 is:

A plane left 30 minutes later than the scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 250 km/h from its usual speed. Find the usual speed of the plane.

If the roots of the equation x² - 8x + k = 0 are such that one root is thrice the other, then the value of k is:

Let's adjust the options.

If x² + kx + 4 = 0 has distinct real roots, find the range of k.

The sum of the ages of a father and his son is 45 years. The product of their ages is 126. Find their present ages.

If alpha and beta are the roots of x² + px + q = 0, then the value of 1/alpha² + 1/beta² is:

The hypotenuse of a right triangle is 25 cm. The difference between the other two sides is 5 cm. Find the lengths of these sides.

If one root of the equation x² - (k+1)x + 10 = 0 is 2, find the other root.

For what values of p does the quadratic equation x² - 2px + 1 = 0 have no real roots?

Revised Question 49: The roots of the quadratic equation x² + ax + b = 0 are 1 and -2. Find the roots of the equation x² + (a+b)x + b = 0.

The roots of the quadratic equation x² + ax + b = 0 are 2 and 3. Find the roots of the equation x² + (a+b)x + b = 0.

Let's recheck the calculation of k. k/2 = 1. So k=2. Let's adjust the options.

If alpha and beta are the roots of the equation 2x² + 5x + k = 0, and alpha² + beta² + alpha*beta = 21/4, find the value of k.

Revised Question 47: The roots of the equation x² - (k+1)x + (k+4) = 0 are real and equal. Find the value of k.

The roots of the equation x² + (k-3)x + (k-2) = 0 are real and equal. Find the value of k.

Revised Question 46: A piece of cloth costs Rs 30. If the piece were 1 m longer and each metre costs Re 1 less, the cost would remain unchanged. What is the length of the piece?

A piece of cloth costs Rs 35. If the piece were 4 m longer and each metre costs Re 1 less, the cost would remain unchanged. What is the length of the piece?

Let me make the options match my calculations.

If one root of the quadratic equation 2x² + kx - 6 = 0 is 2, find the other root.

The value of sqrt(6 + sqrt(6 + sqrt(6 + ...))) is:

Revised Question 43: If the equation (m-1)x² - 2(m-1)x + 1 = 0 has equal roots, then find m.

If the equation (m²+1)x² - 2(m+1)x + 1 = 0 has equal roots, then find m.

The roots of the equation 2x² + 3x + 1 = 0 are p and q. Find the quadratic equation whose roots are 1/p and 1/q.

If alpha and beta are roots of x² - 3x - 2 = 0, then the value of 1/alpha + 1/beta is:

Revised Question 40: A man buys a number of pens for Rs. 60. If he had bought 3 more pens for the same amount, each pen would have cost him Re. 1 less. How many pens did he buy?

A man buys a number of pens for Rs. 180. If he had bought 3 more pens for the same amount, each pen would have cost him Re. 1 less. How many pens did he buy?

Let's make the target roots easier to match the given options. Revised Question 39: If alpha and beta are the roots of the equation x² - 5x + 6 = 0, then the equation whose roots are 1/alpha and 1/beta is:

If alpha and beta are the roots of the equation x² + 7x + 12 = 0, then the equation whose roots are (alpha+beta)² and (alpha-beta)² is:

For what value of p does the quadratic equation 4x² - 3px + 9 = 0 have exactly one real root?

The sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, find the sides of the two squares.

Revised Question 36: If alpha and beta are the roots of 3x² - 6x + 4 = 0, find the value of (alpha/beta) + (beta/alpha).

If alpha and beta are the roots of 3x² - 6x + 4 = 0, find the value of (alpha/beta) + (beta/alpha).

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. What are the distances from the gates?

What is the number of real roots for the equation x² + 4x + 5 = 0?

If a and b are the roots of the equation x² - px + q = 0, then the value of a² + b² is:

Revised Question 32: If the equation (1+m)x² - 2(1+3m)x + (1+8m) = 0 has equal roots, find the value of m.

If the equation (1+m)x² - 2(1+3m)x + (1+8m) = 0 has equal roots, find the value of m.

The sum of a number and its reciprocal is 13/6. Find the number.

Let's adjust the options for the calculated k=-9/4.

If x = -1/2 is a root of the equation 3x² + 2kx - 3 = 0, find the value of k.

Revised Question 29: A farmer has 40 m of fencing material. He wants to enclose a rectangular plot of land using an existing wall as one side of the plot. If the area of the plot is 200 m², what are the dimensions of the plot?

A farmer wishes to fence a rectangular area of 100 m². He has 30 m of fencing material. Find the dimensions of the rectangular area.

Find the value of k such that the quadratic equation (k-12)x² + 2(k-12)x + 2 = 0 has distinct real roots.

If -5 is a root of the quadratic equation 2x² + px - 15 = 0, and the quadratic equation p(x² + x) + k = 0 has equal roots, then find the value of k.

The roots of the equation x² - 6x + k = 0 are real and equal. Find the value of k.

A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the original speed of the train.

If alpha and beta are the roots of the equation x² - x + 1 = 0, then the equation whose roots are alpha³ and beta³ is:

If the sum of the squares of two consecutive even integers is 340, what are the integers?

What is the condition for the roots of the equation ax² + bx + c = 0 to be real and distinct?

If x = 1 is a common root of ax² + ax + 3 = 0 and x² + x + b = 0, then find the value of ab.

The product of Ram's age (in years) five years ago and his age (in years) nine years later is 15. Determine Ram's present age.

If alpha and beta are the roots of the equation x² + x + 1 = 0, find the value of alpha² + beta².

The roots of the quadratic equation x² - 2(a+b)x + 2(a² + b²) = 0 are real. What can be concluded about a and b?

If the roots of the equation (c² - ab)x² - 2(a² - bc)x + (b² - ac) = 0 are equal, then which condition must be true?

A rectangular park is to be designed whose breadth is 5m less than its length. Its area is to be numerically equal to the perimeter of a square park whose side is equal to the length of the rectangular park. Find the length of the rectangular park.

If alpha and beta are the roots of x² - 5x + 6 = 0, then find the equation whose roots are 1/alpha and 1/beta.

If the roots of the equation x² + 2cx + ab = 0 are real and unequal, and the roots of x² - 2(a+b)x + a² + b² + 2c² = 0 are equal, then the value of c is:

For what value of m does the quadratic equation (4-m)x² + (2m+4)x + (8m+1) = 0 have no real roots?

A two-digit number is such that the product of its digits is 14. When 45 is added to the number, the digits interchange their places. Find the number.

If the sum of the roots of the quadratic equation (k+1)x² - (2k-1)x + 3k = 0 is 1, then the product of the roots is:

If alpha and beta are the roots of x² - p(x+1) - c = 0, then find the value of (alpha+1)(beta+1).

If alpha and beta are the roots of the equation 2x² - 3x + 5 = 0, find the value of (alpha + beta)².

If the roots of the equation (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 are equal, then which of the following is true?

If one root of the equation ax² + bx + c = 0 is the reciprocal of the other, then which of the following is true?

If x = 2/3 and x = -3 are the roots of the quadratic equation ax² + 7x + b = 0, find the values of a and b.

A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

If one root of the quadratic equation x² + px + q = 0 is twice the other, then which of the following is true?

For what value of k does the quadratic equation kx(x-2) + 6 = 0 have two real equal roots?

Which of the following equations is NOT a quadratic equation?