10th Class Mathematics Unit 2 Quadratic Equations Numericals with Solutions

2.1.1.(i) Write the following quadratic equation in standard form: 3x - 1 = 2x² (ii) Write the following quadratic equation in standard form: 2x(x + 1) = 4(2x + 3) (iii) Write the following quadratic equation in standard form: 2x² - 4x = 4x + 7 (iv) Write the following quadratic equation in standard form: 4(3x - 2) = 9x² (v) Write the following quadratic equation in standard form: 2x + 1/x = 5 - 1/x, x ≠ 0 (vi) Write the following quadratic equation in standard form: (6x+6)/(20-x) = 1/x, x ≠ 0, 20

Given

(i) 3x - 1 = 2x²

(i) 2x² - 3x + 1 = 0

Result

(i) Standard form 2x² - 3x + 1 = 0 quad (a=2, b=-3, c=1)

Given

(ii) 2x(x+1) = 4(2x+3)

(ii) 2x² + 2x = 8x + 12

(ii) 2x² - 6x - 12 = 0 ⇒ x² - 3x - 6 = 0 (divide by 2)

Result

(ii) Standard form x² - 3x - 6 = 0

Given

(iii) 2x² - 4x = 4x + 7

(iii) 2x² - 8x - 7 = 0

Result

(iii) Standard form 2x² - 8x - 7 = 0

Given

(iv) 4(3x-2) = 9x²

(iv) 12x - 8 = 9x²

Result

(iv) Standard form 9x² - 12x + 8 = 0

Given

(v) 2x + 1x = 5 - 1x, x neq 0

(v) 2x + 2x - 5 = 0

(v) Multiply by x 2x² - 5x + 2 = 0

Result

(v) Standard form 2x² - 5x + 2 = 0

Given

(vi) 6x+620-x = 1x, x neq 0,20

(vi) Cross-multiply x(6x+6) = 20-x

(vi) 6x²+7x-20=0

Result

(vi) Standard form 6x²+7x-20=0

2.1.2.(i) Solve x² - x - 6 = 0 by factorization. (ii) Solve x² + 3x - 28 = 0 by factorization. (iii) Solve 6x² + 13x - 5 = 0 by factorization. (iv) Solve x² - (3/2)x = 9/2 by factorization. (v) Solve (3x-8)/(x-2) = (5x-2)/(x+5), x ≠ 2, -5 by factorization. (vi) Solve 1/(x-1) - 1/(x+3) = 1/35, x ≠ 1, -3 by factorization.

Given

(i) x²-x-6=0

(i) Numbers with product -6, sum -1 -3, 2

(i) (x-3)(x+2)=0

Result

(i) x = 3, -2

Given

(ii) x²+3x-28=0

(ii) Numbers with product -28, sum 3 7, -4

(ii) (x+7)(x-4)=0

Result

(ii) x = -7, 4

Given

(iii) 6x²+13x-5=0

(iii) Split middle term (product 6(-5)=-30, sum 13 -> 15,-2) 6x²+15x-2x-5=0

(iii) 3x(2x+5)-1(2x+5)=0 ⇒ (3x-1)(2x+5)=0

Result

(iii) x = 13, -52

Given

(iv) x² - 32x = 92

(iv) Multiply by 2 2x²-3x-9=0

(iv) Split middle term (2(-9)=-18, sum -3 -> -6,3) 2x(x-3)+3(x-3)=0 ⇒ (2x+3)(x-3)=0

Result

(iv) x = -32, 3

Given

(v) 3x-8x-2 = 5x-2x+5

(v) Cross-multiply (3x-8)(x+5) = (5x-2)(x-2)

(v) 2x²-19x+44=0 ⇒ (2x-11)(x-4)=0

Result

(v) x = 112, 4

Given

(vi) 1x-1 - 1x+3 = 135

(vi) Take LCM = 35(x-1)(x+3) 35(x+3)-35(x-1) = (x-1)(x+3)

(vi) 140 = x²+2x-3 ⇒ x²+2x-143=0 ⇒ (x+13)(x-11)=0

Result

(vi) x = -13, 11

2.1.3.(i) Solve 2x² + 5x + 2 = 0 by completing the square. (ii) Solve x² + x = 42 by completing the square. (iii) Solve 12x² + 7x = 12 by completing the square. (iv) Solve (x+3)/(2x-7) = (2x-1)/(x-3), x ≠ 7/2, 3 by completing the square. (v) Solve 1/(1+x) - 1/(3-x) = 6/35, x ≠ -1, 3 by completing the square. (vi) Solve (3x-1)/(4x+7) = 1 - 6/(x+7), x ≠ -7/4, -7 by completing the square.

Given

(i) 2x²+5x+2=0

(i) Divide by 2 x²+52x = -1

(i) Complete the square left(x+54right)² = 916 ⇒ x+54= ± 34

Result

(i) x = -12, -2

Given

(ii) x²+x=42

(ii) Add (1/2)² to both sides x²+x+14 = 42+14 ⇒ left(x+12right)²=1694

(ii) x+12 = ± 132

Result

(ii) x = 6, -7

Given

(iii) 12x²+7x=12

(iii) Divide by 12 x²+712x = 1

(iii) Complete the square left(x+724right)²=625576 ⇒ x+724= ± 2524

Result

(iii) x = 34, -43

Given

(iv) x+32x-7 = 2x-1x-3

(iv) Cross-multiply (x+3)(x-3) = (2x-1)(2x-7)

(iv) x²-9 = 4x²-16x-2x+7 ⇒ 0 = 3x²-18x+16

(2x-1)(2x-7) expands to 4x^2-16x+7 (cross terms -14x and -2x sum to -16x, not -18x). The correct equation is 3x^2-16x+16=0, giving x=4 or x=4/3.

(iv) Divide by 3, complete the square (x-3)² = 113 ⇒ x = 3 ± frac{sqrt{33}{3}

Result

(iv) x = 3+frac{sqrt{33}{3}, 3-frac{sqrt{33}{3}

Given

(v) 11+x - 13-x = 635

(v) LCM = (1+x)(3-x) 2-2x(1+x)(3-x) = 635

(v) 35(2-2x) = 6(1+x)(3-x)

(1+x)(3-x) = 3+2x-x^2, not 3-x^2 (the +2x term was dropped). The correct equation is 6x^2-82x+52=0, i.e. 3x^2-41x+26=0, giving x=13 or x=2/3.

(v) 70-70x=18-6x² ⇒ 6x²-70x-52=0 ⇒ 3x²-35x-26=0 ⇒ (x-13)(3x+2)=0

Result

(v) x = 13, -23

Given

(vi) 3x-14x+7 = 1 - 6x+7

(vi) Multiply both sides by (4x+7)(x+7) (3x-1)(x+7) = (x+7)(x+7) - 6(4x+7)

The right side should be (4x+7)(x+7) - 6(4x+7), not (x+7)(x+7) - 6(4x+7); the factor (4x+7) was mistakenly replaced by (x+7). Correct expansion gives x^2-9x+14=0, i.e. x=7 or x=2.

(vi) 3x²+20x-7 = x²-10x+7 ⇒ 2x²+30x-14=0 ⇒ x²+15x-7=0

(vi) Complete the square left(x+152right)² = 2534

Result

(vi) x = frac{-15+sqrt{253}{2}, frac{-15-sqrt{253}{2}

2.1.4.(i) Use the quadratic formula to solve 2x² - 5x + 3 = 0. (ii) Use the quadratic formula to solve 2x² - 7x - 15 = 0. (iii) Use the quadratic formula to solve 2x² + 7x = 15. (iv) Use the quadratic formula to solve x² + 11 = 7x. (v) Use the quadratic formula to solve (x+4)/(x-4) + (x-2)/(x-3) = 6 1/3, x ≠ 4, 3. (vi) Use the quadratic formula to solve (3x-3)/(x+1) = (2x-1)/(x-1), x ≠ -1, 1.

Given

(i) 2x²-5x+3=0, a=2, b=-5, c=3

Formula

(i) Quadratic formula x = frac{-b ± sqrt{b²-4ac}{2a}

Working

(i) x = frac{5 ± sqrt{25-24}{4} = 5 ± 14

Result

(i) x = 32, 1

Given

(ii) 2x²-7x-15=0, a=2, b=-7, c=-15

Working

(ii) x = frac{7 ± sqrt{49+120}{4} = 7 ± 134

Result

(ii) x = 5, -32

Given

(iii) 2x²+7x-15=0, a=2, b=7, c=-15

Working

(iii) x = frac{-7 ± sqrt{49+120}{4} = -7 ± 134

Result

(iii) x = 32, -5

Given

(iv) x²-7x+11=0, a=1, b=-7, c=11

Working

(iv) x = frac{7 ± sqrt{49-44}{2}

Result

(iv) x = 7+sqrt52, 7-sqrt52

Given

(v) x+4x-4 + x-2x-3 = 193

(v) Take LCM, multiply by 3 13x²-118x+240=0

Working

(v) x = frac{118 ± sqrt{13924-12480}{26} = 118 ± 3826

Result

(v) x = 6, 4013

Given

(vi) 3x-3x+1 = 2x-1x-1

(vi) Cross-multiply 3x²-6x+3 = 2x²+x-1 ⇒ x²-7x+4=0

Working

(vi) x = frac{7 ± sqrt{49-16}{2}

Result

(vi) x = frac{7+sqrt{33}{2}, frac{7-sqrt{33}{2}

2.1.5.(i) Solve x² - 3x - 18 = 0 graphically. (ii) Solve x² - 5x - 14 = 0 graphically. (iii) Solve 2x² + 13x + 6 = 0 graphically. (iv) Solve 4x² + 12x - 27 = 0 graphically.

Given

(i) y = x²-3x-18

(i) x-intercepts from graph/table Vertex (1.5, -20.25)

Result

(i) x = -3, 6

Given

(ii) y = x²-5x-14

(ii) x-intercepts from graph/table Vertex (2.5, -20.25)

Result

(ii) x = -2, 7

Given

(iii) y = 2x²+13x+6

(iii) Table and graph table values shown do not satisfy y=2x²+13x+6

The printed table (e.g. y=-4 at x=-5, y=0 at x=-2) does not match y=2x^2+13x+6, and three x-intercepts are not possible for a quadratic. Solving directly: x=(-13\pm\sqrt{121})/4 = -1/2 or -6.

Result

(iii) x = -3, -2, 1

Given

(iv) y = 4x²+12x-27

(iv) Vertex Vertex (-1.5, -36)

Result

(iv) x = -4.5, 1.5

Unit 2: Quadratic Equations — Exercise 2 1