Exercise 4.1 answers: Quadratic Equations

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(i) (x + 1)² = 2(x − 3)

1. Expand the left side: (x + 1)² = x² + 2x + 1.
2. Expand the right side: 2(x − 3) = 2x − 6.
3. Move everything to one side: (x² + 2x + 1) − (2x − 6) = x² + 7.
4. The equation becomes x² + 7 = 0. The x² term survives.

AnswerYes, it is a quadratic equation.

(ii) x² − 2x = (−2) (3 − x)

1. Expand the right side: −2(3 − x) = −6 + 2x.
2. Move everything to one side: (x² − 2x) − (−6 + 2x) = x² − 4x + 6.
3. The equation becomes x² − 4x + 6 = 0. The x² term survives.

AnswerYes, it is a quadratic equation.

(iii) (x − 2)(x + 1) = (x − 1)(x + 3)

1. Expand the left side: (x − 2)(x + 1) = x² − x − 2.
2. Expand the right side: (x − 1)(x + 3) = x² + 2x − 3.
3. Move everything to one side: (x² − x − 2) − (x² + 2x − 3) = −3x + 1.
4. The x² terms cancel, leaving −3x + 1 = 0, which has degree 1.

AnswerNo, it is not a quadratic equation — it is linear.

(iv) (x − 3)(2x +1) = x(x + 5)

1. Expand the left side: (x − 3)(2x + 1) = 2x² − 5x − 3.
2. Expand the right side: x(x + 5) = x² + 5x.
3. Move everything to one side: (2x² − 5x − 3) − (x² + 5x) = x² − 10x − 3.
4. The equation becomes x² − 10x − 3 = 0. The x² term survives.

AnswerYes, it is a quadratic equation.

(v) (2x−1)(x−3) = (x+5)(x−1)

1. Expand the left side: (2x − 1)(x − 3) = 2x² − 7x + 3.
2. Expand the right side: (x + 5)(x − 1) = x² + 4x − 5.
3. Move everything to one side: (2x² − 7x + 3) − (x² + 4x − 5) = x² − 11x + 8.
4. The equation becomes x² − 11x + 8 = 0. The x² term survives.

AnswerYes, it is a quadratic equation.

(vi) x² + 3x + 1 = (x – 2)²

1. Expand the right side: (x − 2)² = x² − 4x + 4.
2. Move everything to one side: (x² + 3x + 1) − (x² − 4x + 4) = 7x − 3.
3. The x² terms cancel, leaving 7x − 3 = 0, which has degree 1.

AnswerNo, it is not a quadratic equation — it is linear.

(vii) (x + 2)³ = 2x (x² – 1)

1. Expand the left side: (x + 2)³ = x³ + 6x² + 12x + 8.
2. Expand the right side: 2x(x² − 1) = 2x³ − 2x.
3. Move everything to one side: (x³ + 6x² + 12x + 8) − (2x³ − 2x) = −x³ + 6x² + 14x + 8.
4. The highest surviving power is 3, so this is a cubic equation, not a quadratic one.

AnswerNo, it is not a quadratic equation.

(viii) x³ – 4x² - x + 1 = (x – 2)³

1. Expand the right side: (x − 2)³ = x³ − 6x² + 12x − 8.
2. Move everything to one side: (x³ − 4x² − x + 1) − (x³ − 6x² + 12x − 8) = 2x² − 13x + 9.
3. The x³ terms cancel, leaving 2x² − 13x + 9 = 0, which has degree 2.

AnswerYes, it is a quadratic equation.

Watch this explained “Eight to sort”, 7:53 into The shape an equation has to have before it counts as quadratic

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(i) The area of a rectangular plot is 528 m².

1. Let the breadth of the plot be b metres.
2. The length is one more than twice the breadth, so length = 2b + 1.
3. Area = breadth × length = b(2b + 1) = 2b² + b, and this equals 528.
4. Move 528 across: 2b² + b − 528 = 0.

Answer2b² + b − 528 = 0, where b is the breadth in metres.

(ii) The product of two consecutive positive integers is 306.

1. Let the smaller integer be x.
2. The next consecutive integer is x + 1.
3. Their product is 306, so x(x + 1) = 306.
4. Move 306 across: x² + x − 306 = 0.

Answerx² + x − 306 = 0, where x is the smaller integer.

(iii) Rohan’s mother is 26 years older than him.

1. Let Rohan's present age be x years, so his mother's present age is x + 26.
2. Three years from now their ages are x + 3 and x + 29.
3. The product of their ages then is 360: (x + 3)(x + 29) = 360.
4. Expand and move 360 across: x² + 32x + 87 − 360 = 0, i.e. x² + 32x − 273 = 0.

Answerx² + 32x − 273 = 0, where x is Rohan's present age in years.

(iv) A train travels a distance of 480 km at a uniform speed.

1. Let the speed of the train be x km/h. Time = distance ÷ speed, so 480 km takes 480/x hours.
2. At (x − 8) km/h the same 480 km takes 480/(x − 8) hours. The slower journey takes 3 hours more.
3. So 480/(x − 8) − 480/x = 3.
4. Multiply both sides by x(x − 8) to clear the fractions: 480x − 480(x − 8) = 3x(x − 8).
5. The left side is 480x − 480x + 3840 = 3840, and the right side is 3x² − 24x. So 3x² − 24x − 3840 = 0.
6. Divide every term by 3: x² − 8x − 1280 = 0.

Answerx² − 8x − 1280 = 0, where x is the speed of the train in km/h.

Watch this explained “Two more, and you have seen their shapes”, 7:50 into Turning a described situation into an equation of that shape