Quadratic Equations | Exercise 4.3

Question 1

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x2−3x+5=02x^2 - 3x + 5 = 0

(ii) 3x2−43x+4=03x^2 - 4\sqrt{3}x + 4 = 0

(iii) 2x2−6x+3=02x^2 - 6x + 3 = 0

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Solution
Understand the Question

To find the nature of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, we calculate the discriminant D=b2−4acD = b^2 - 4ac:

  • D>0D > 0: Two distinct real roots exist.
  • D=0D = 0: Two equal real roots exist.
  • D<0D < 0: No real roots exist.

When real roots exist, they are given by the quadratic formula: x=−b±D2ax = \dfrac{-b \pm \sqrt{D}}{2a}

(i) 2x2−3x+5=02x^2 - 3x + 5 = 0

Step 1 · Calculate Discriminant and Determine Nature of Roots

Comparing 2x2−3x+5=02x^2 - 3x + 5 = 0 with ax2+bx+c=0ax^2 + bx + c = 0: a=2,b=−3,c=5a = 2, \quad b = -3, \quad c = 5

Calculate the discriminant DD:

D=b2−4ac=(−3)2−4×2×5=9−40=−31\begin{aligned} D &= b^2 - 4ac \\ &= (-3)^2 - 4 \times 2 \times 5 \\ &= 9 - 40 \\ &= -31 \end{aligned}

Since D<0D < 0, there are no real roots for this equation.

Answer

(i) No real roots

(ii) 3x2−43x+4=03x^2 - 4\sqrt{3}x + 4 = 0

Step 1 · Calculate Discriminant and Determine Nature of Roots

Comparing 3x2−43x+4=03x^2 - 4\sqrt{3}x + 4 = 0 with ax2+bx+c=0ax^2 + bx + c = 0: a=3,b=−43,c=4a = 3, \quad b = -4\sqrt{3}, \quad c = 4

Calculate the discriminant DD:

D=b2−4ac=(−43)2−4×3×4=(16×3)−48=48−48=0\begin{aligned} D &= b^2 - 4ac \\ &= (-4\sqrt{3})^2 - 4 \times 3 \times 4 \\ &= (16 \times 3) - 48 \\ &= 48 - 48 \\ &= 0 \end{aligned}

Since D=0D = 0, the roots are real and equal.

Step 2 · Find the Roots

Using the quadratic formula:

x=−b±D2a=−(−43)±02×3=436=233\begin{aligned} x &= \dfrac{-b \pm \sqrt{D}}{2a} \\[0.6em] &= \dfrac{-(-4\sqrt{3}) \pm \sqrt{0}}{2 \times 3} \\[0.6em] &= \dfrac{4\sqrt{3}}{6} \\[0.6em] &= \dfrac{2\sqrt{3}}{3} \end{aligned}

Therefore, both roots are equal to 233\dfrac{2\sqrt{3}}{3} (or 23\dfrac{2}{\sqrt{3}}).

Answer

(ii) Real and equal roots: 233,233\dfrac{2\sqrt{3}}{3}, \dfrac{2\sqrt{3}}{3}

(iii) 2x2−6x+3=02x^2 - 6x + 3 = 0

Step 1 · Calculate Discriminant and Determine Nature of Roots

Comparing 2x2−6x+3=02x^2 - 6x + 3 = 0 with ax2+bx+c=0ax^2 + bx + c = 0: a=2,b=−6,c=3a = 2, \quad b = -6, \quad c = 3

Calculate the discriminant DD:

D=b2−4ac=(−6)2−4×2×3=36−24=12\begin{aligned} D &= b^2 - 4ac \\ &= (-6)^2 - 4 \times 2 \times 3 \\ &= 36 - 24 \\ &= 12 \end{aligned}

Since D>0D > 0, the roots are real and distinct.

Step 2 · Find the Roots

Using the quadratic formula:

x=−b±D2a=−(−6)±122×2=6±234=2(3±3)4=3±32\begin{aligned} x &= \dfrac{-b \pm \sqrt{D}}{2a} \\[0.6em] &= \dfrac{-(-6) \pm \sqrt{12}}{2 \times 2} \\[0.6em] &= \dfrac{6 \pm 2\sqrt{3}}{4} \\[0.6em] &= \dfrac{2(3 \pm \sqrt{3})}{4} \\[0.6em] &= \dfrac{3 \pm \sqrt{3}}{2} \end{aligned}
Answer

(iii) Real and distinct roots: 3+32,3−32\dfrac{3 + \sqrt{3}}{2}, \dfrac{3 - \sqrt{3}}{2}

Common Mistakes
  • Sign Error in −b-b: Forgetting that when bb is negative (e.g., b=−6b = -6), −b=−(−6)=+6-b = -(-6) = +6.
  • Squaring Negative Surds: Miscalculating (−43)2(-4\sqrt{3})^2. Remember that (−4)2×(3)2=16×3=48(-4)^2 \times (\sqrt{3})^2 = 16 \times 3 = 48.
  • Stating Only One Root for D=0D = 0: A quadratic equation always has two roots. When D=0D = 0, both roots are equal and should be stated twice (e.g., 233,233\dfrac{2\sqrt{3}}{3}, \dfrac{2\sqrt{3}}{3}).
  • Incorrect Simplification: Forgetting to factor out the common term from the numerator before cancelling with the denominator in 6±234=2(3±3)4=3±32\dfrac{6 \pm 2\sqrt{3}}{4} = \dfrac{2(3 \pm \sqrt{3})}{4} = \dfrac{3 \pm \sqrt{3}}{2}.

More questions in Exercise 4.3

Q1

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x2−3x+5=02x^2 - 3x + 5 = 0

(ii) 3x2−43x+4=03x^2 - 4\sqrt{3}x + 4 = 0

(iii) 2x2−6x+3=02x^2 - 6x + 3 = 0

Q2

Find the values of kk for each of the following quadratic equations, so that they have two equal roots.

(i) 2x2+kx+3=02x^2 + kx + 3 = 0

(ii) kx(x−2)+6=0kx (x - 2) + 6 = 0

Q3

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2800 \text{ m}^2? If so, find its length and breadth.

Q4

Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 2020 years. Four years ago, the product of their ages in years was 4848.

Q5

Is it possible to design a rectangular park of perimeter 80 m80\text{ m} and area 400 m2400\text{ m}^2? If so, find its length and breadth.

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