- A(x+1)² = 2(x-3)
- Bx(x+1) + 8 = (x+2)(x-2)
- Cx(2x+3) = x² + 1
- D(x+2)³ - x³ = 6x² + 12x + 8
Simplifying the options: A simplifies to x² + 7 = 0 (quadratic). B simplifies to x² + x + 8 = x² - 4, which reduces to x + 12 = 0 (linear). C simplifies to x² + 3x - 1 = 0 (quadratic). D simplifies to 6x² + 12x + 8 = 6x² + 12x + 8, which is an identity (0=0), not an equation in x with a non-zero quadratic term.
More Questions on Quadratic Equations
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.
→ 2The roots of the equation x² - (k+1)x + (k+4) = 0 are real and equal. Find the value of k.
→ 3A 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?
→ 4If the equation (m-1)x² - 2(m-1)x + 1 = 0 has equal roots, then find m.
→ 5A 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?
→ 6If 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:
→ 7A 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?
→ 8If 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:
→ 9The 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.
→ 10For what value of m does the quadratic equation (m-1)x² + 2(m-1)x + 1 = 0 have exactly one real root?
→ 11Revised Question 58: If alpha and beta are the roots of the equation x² - 7x + 10 = 0, find the value of alpha³ + beta³.
→ 12If alpha and beta are the roots of the equation x² - 7x + 10 = 0, find the value of alpha³ + beta³.
→ 13The product of two consecutive positive integers is 306. Find the integers.
→ 14Revised Question 56: If a is a root of the equation x² - x - 1 = 0, then the value of a³ - 2a + 1 is:
→ 15If a is a root of the equation x² - x - 1 = 0, then the value of a³ - 2a + 1 is:
→ 16A 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.
→ 17If the roots of the equation x² - 8x + k = 0 are such that one root is thrice the other, then the value of k is:
→ 18Let's adjust the options.
→ 19If x² + kx + 4 = 0 has distinct real roots, find the range of k.
→ 20The sum of the ages of a father and his son is 45 years. The product of their ages is 126. Find their present ages.
→ 21If alpha and beta are the roots of x² + px + q = 0, then the value of 1/alpha² + 1/beta² is:
→ 22The hypotenuse of a right triangle is 25 cm. The difference between the other two sides is 5 cm. Find the lengths of these sides.
→ 23If one root of the equation x² - (k+1)x + 10 = 0 is 2, find the other root.
→ 24For what values of p does the quadratic equation x² - 2px + 1 = 0 have no real roots?
→ 25Revised 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.
→ 26The 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.
→ 27Let's recheck the calculation of k. k/2 = 1. So k=2. Let's adjust the options.
→ 28If 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.
→ 29Revised Question 47: The roots of the equation x² - (k+1)x + (k+4) = 0 are real and equal. Find the value of k.
→ 30The roots of the equation x² + (k-3)x + (k-2) = 0 are real and equal. Find the value of k.
→ 31Revised 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?
→ 32A 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?
→ 33Let me make the options match my calculations.
→ 34If one root of the quadratic equation 2x² + kx - 6 = 0 is 2, find the other root.
→ 35The value of sqrt(6 + sqrt(6 + sqrt(6 + ...))) is:
→ 36Revised Question 43: If the equation (m-1)x² - 2(m-1)x + 1 = 0 has equal roots, then find m.
→ 37If the equation (m²+1)x² - 2(m+1)x + 1 = 0 has equal roots, then find m.
→ 38The roots of the equation 2x² + 3x + 1 = 0 are p and q. Find the quadratic equation whose roots are 1/p and 1/q.
→ 39If alpha and beta are roots of x² - 3x - 2 = 0, then the value of 1/alpha + 1/beta is:
→ 40Revised 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?
→ 41A 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?
→ 42Let'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:
→ 43If 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:
→ 44For what value of p does the quadratic equation 4x² - 3px + 9 = 0 have exactly one real root?
→ 45The 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.
→ 46Revised Question 36: If alpha and beta are the roots of 3x² - 6x + 4 = 0, find the value of (alpha/beta) + (beta/alpha).
→ 47If alpha and beta are the roots of 3x² - 6x + 4 = 0, find the value of (alpha/beta) + (beta/alpha).
→ 48A 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?
→ 49What is the number of real roots for the equation x² + 4x + 5 = 0?
→ 50If a and b are the roots of the equation x² - px + q = 0, then the value of a² + b² is:
→ 51Revised Question 32: If the equation (1+m)x² - 2(1+3m)x + (1+8m) = 0 has equal roots, find the value of m.
→ 52If the equation (1+m)x² - 2(1+3m)x + (1+8m) = 0 has equal roots, find the value of m.
→ 53The sum of a number and its reciprocal is 13/6. Find the number.
→ 54Let's adjust the options for the calculated k=-9/4.
→ 55If x = -1/2 is a root of the equation 3x² + 2kx - 3 = 0, find the value of k.
→ 56Revised 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?
→ 57A farmer wishes to fence a rectangular area of 100 m². He has 30 m of fencing material. Find the dimensions of the rectangular area.
→ 58Find the value of k such that the quadratic equation (k-12)x² + 2(k-12)x + 2 = 0 has distinct real roots.
→ 59If -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.
→ 60The roots of the equation x² - 6x + k = 0 are real and equal. Find the value of k.
→ 61A 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.
→ 62If alpha and beta are the roots of the equation x² - x + 1 = 0, then the equation whose roots are alpha³ and beta³ is:
→ 63If the sum of the squares of two consecutive even integers is 340, what are the integers?
→ 64What is the condition for the roots of the equation ax² + bx + c = 0 to be real and distinct?
→ 65If x = 1 is a common root of ax² + ax + 3 = 0 and x² + x + b = 0, then find the value of ab.
→ 66The 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.
→ 67If alpha and beta are the roots of the equation x² + x + 1 = 0, find the value of alpha² + beta².
→ 68The roots of the quadratic equation x² - 2(a+b)x + 2(a² + b²) = 0 are real. What can be concluded about a and b?
→ 69If the roots of the equation (c² - ab)x² - 2(a² - bc)x + (b² - ac) = 0 are equal, then which condition must be true?
→ 70A 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.
→ 71If alpha and beta are the roots of x² - 5x + 6 = 0, then find the equation whose roots are 1/alpha and 1/beta.
→ 72If 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:
→ 73For what value of m does the quadratic equation (4-m)x² + (2m+4)x + (8m+1) = 0 have no real roots?
→ 74A 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.
→ 75If 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:
→ 76If alpha and beta are the roots of x² - p(x+1) - c = 0, then find the value of (alpha+1)(beta+1).
→ 77If alpha and beta are the roots of the equation 2x² - 3x + 5 = 0, find the value of (alpha + beta)².
→ 78If the roots of the equation (a² + b²)x² - 2(ac + bd)x + (c² + d²) = 0 are equal, then which of the following is true?
→ 79If one root of the equation ax² + bx + c = 0 is the reciprocal of the other, then which of the following is true?
→ 80For what values of p does the quadratic equation x² + px + 1 = 0 have real roots?
→ 81If x = 2/3 and x = -3 are the roots of the quadratic equation ax² + 7x + b = 0, find the values of a and b.
→ 82A 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.
→ 83If one root of the quadratic equation x² + px + q = 0 is twice the other, then which of the following is true?
→ 84For what value of k does the quadratic equation kx(x-2) + 6 = 0 have two real equal roots?
→ 85Which of the following equations is NOT a quadratic equation?
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