Quadratic equations
Class 10 Maths · Quadratic equations · 30 questions with answers
Quadratics is the chapter Class 8 factorisation was built for, and it repays that work directly. Three things are asked: solve it, say what kind of roots it has, and turn a described situation into an equation. The third is where marks are actually lost, and it is lost in the first line rather than in the algebra.
- Solves a quadratic by splitting the middle term
- Applies the quadratic formula correctly, including the sign of b
- Uses the discriminant to state the nature of the roots
- Finds an unknown coefficient given a condition on the roots
- Forms a quadratic equation from a described situation
Exercise 1 of 3 · 10 questions
Solve by factorisation.
Write the answer · Warm-up
Here’s one done for you
The last one looks alarming and works exactly like the others. The product to aim for is (first coefficient) × (last term) = √2 × 5√2 = 10, and the sum is 7, giving 5 and 2. Split the middle term as 5x + 2x and group. Surds in the coefficients change nothing about the method - only about how the arithmetic looks, and children who realise that stop being frightened by these.
- 1)x² - 5x + 6 = 0
- 2)2x² + x - 6 = 0
- 3)x² - 3x - 10 = 0
- 4)6x² - x - 2 = 0
- 5)√2x² + 7x + 5√2 = 0
- 6)x² - 7x + 12 = 0
- 7)3x² - 10x + 8 = 0
- 8)x² + 2x - 8 = 0
- 9)2x² - 5x - 3 = 0
- 10)x² - 16 = 0
Answers
- 1) (x - 2)(x - 3) = 0, so x = 2 or x = 3.
- 2) (2x - 3)(x + 2) = 0, so x = 3/2 or x = -2. Split as 2x² + 4x - 3x - 6.
- 3) (x - 5)(x + 2) = 0, so x = 5 or x = -2.
- 4) (3x - 2)(2x + 1) = 0, so x = 2/3 or x = -1/2. Split as 6x² - 4x + 3x - 2.
- 5) (√2x + 5)(x + √2) = 0, so x = -5/√2 or x = -√2. The product is √2 × 5√2 = 10 and the sum is 7, so split as 5x + 2x.
- 6) (x - 3)(x - 4) = 0, so x = 3 or x = 4.
- 7) (3x - 4)(x - 2) = 0, so x = 4/3 or x = 2.
- 8) (x + 4)(x - 2) = 0, so x = -4 or x = 2.
- 9) (2x + 1)(x - 3) = 0, so x = -1/2 or x = 3.
- 10) (x + 4)(x - 4) = 0, so x = 4 or x = -4.
Exercise 2 of 3 · 10 questions
Solve using the quadratic formula, then state the nature of the roots.
Write the answer · Guided
Here’s one done for you
For 2x² - 7x + 3, note that b = -7 and not 7, so -b is +7. That sign is the commonest error in the whole chapter, and it survives because the answer still looks like a number. Write a, b and c on their own line before substituting - three seconds, and it removes the error entirely.
- 1)2x² - 7x + 3 = 0
- 2)3x² - 5x + 2 = 0
- 3)x² + 4x - 5 = 0
- 4)2x² - 4x + 3 = 0
- 5)x² - 6x + 9 = 0
- 6)5x² - 6x + 1 = 0
- 7)4x² - 4x + 1 = 0
- 8)x² - 4x + 4 = 0
- 9)3x² + 2x + 5 = 0
- 10)2x² + 5x - 3 = 0
Answers
- 1) D = 49 - 24 = 25, so x = (7 ± 5)/4, giving x = 3 or x = 1/2. Two distinct real roots.
- 2) D = 25 - 24 = 1, so x = (5 ± 1)/6, giving x = 1 or x = 2/3. Two distinct real roots.
- 3) D = 16 + 20 = 36, so x = (-4 ± 6)/2, giving x = 1 or x = -5. Two distinct real roots.
- 4) D = 16 - 24 = -8, which is negative. No real roots.
- 5) D = 36 - 36 = 0, so x = 3 is a repeated root. Two equal real roots.
- 6) D = 36 - 20 = 16, so x = (6 ± 4)/10, giving x = 1 or x = 1/5. Two distinct real roots.
- 7) D = 16 - 16 = 0, so x = 4/8 = 1/2 twice. Two equal real roots.
- 8) D = 16 - 16 = 0, so x = 2 twice. Two equal real roots.
- 9) D = 4 - 60 = -56, which is negative, so there are no real roots.
- 10) D = 25 + 24 = 49, so x = (-5 ± 7)/4, giving x = 1/2 or x = -3. Two distinct real roots.
Exercise 3 of 3 · 10 questions
Form the equation and solve it. Check the answer against the question.
Write the answer · Practice
Here’s one done for you
The train question is the one worth studying, because the algebra is easy and the first line is not. Time equals distance over speed, so the two journey times are 360/x and 360/(x + 5), and the faster one takes an hour less - which gives 360/x - 360/(x + 5) = 1. Once that line is written the rest is routine. In every word problem here, the marks are decided by whether the child can write that first line, so it is worth practising the setting-up separately from the solving.
- 1)Find k so that kx² - 2kx + 6 = 0 has two equal roots.
- 2)The sum of a number and its reciprocal is 10/3. Find the number.
- 3)The product of two consecutive positive integers is 306. Find them.
- 4)A rectangular plot has an area of 528 m², and its length is one metre more than twice its breadth. Find both.
- 5)A train covers 360 km at a uniform speed. Had the speed been 5 km/h more, it would have taken one hour less. Find the speed.
- 6)Find the value of k for which x² + kx + 9 = 0 has equal roots.
- 7)The sum of the squares of two consecutive positive integers is 313. Find them.
- 8)The perimeter of a rectangle is 28 m and its area is 48 m². Find its sides.
- 9)A two-digit number is such that the product of its digits is 18, and when 27 is subtracted the digits interchange. Find the number.
- 10)The difference of two numbers is 3 and their product is 88. Find them.
Answers
- 1) Equal roots require D = 0, so 4k² - 24k = 0, giving 4k(k - 6) = 0 and k = 0 or 6. k = 0 makes it non-quadratic, so k = 6.
- 2) x + 1/x = 10/3 gives 3x² - 10x + 3 = 0. D = 64, so x = (10 ± 8)/6, giving x = 3 or x = 1/3.
- 3) n(n + 1) = 306 gives n² + n - 306 = 0. D = 1225 = 35², so n = 17 (rejecting the negative root). The integers are 17 and 18.
- 4) b(2b + 1) = 528 gives 2b² + b - 528 = 0. D = 4225 = 65², so b = 16 and the length is 33 m.
- 5) 360/x - 360/(x + 5) = 1 gives x² + 5x - 1800 = 0. D = 7225 = 85², so x = 40. The speed is 40 km/h.
- 6) Equal roots need D = 0, so k² - 36 = 0 and k = ±6.
- 7) n² + (n + 1)² = 313 gives 2n² + 2n - 312 = 0, so n² + n - 156 = 0 and (n + 13)(n - 12) = 0. Taking the positive value, the integers are 12 and 13.
- 8) Let the sides be l and 14 - l. Then l(14 - l) = 48, so l² - 14l + 48 = 0 and (l - 6)(l - 8) = 0. The sides are 6 m and 8 m.
- 9) Let the digits be x and y with xy = 18 and 10x + y - 27 = 10y + x, so x - y = 3. Then x = 6 and y = 3, giving 63. Check: 63 - 27 = 36.
- 10) Let them be n and n + 3. Then n(n + 3) = 88, so n² + 3n - 88 = 0 and (n + 11)(n - 8) = 0. The numbers are 8 and 11 (or -11 and -8).
While your child works
- Write a, b and c on their own line before using the formula. The sign of b is the commonest error in the chapter.
- Reject impossible roots explicitly - a negative length or a negative speed. Saying why earns the mark.
- In word problems, practise writing the first line without solving. That is where the marks actually are.