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NCERT Exemplar · Class 10 Mathematics Quadratic Equations

28 questions · 28 still being checked

EXERCISE 4.3 1–2 (part 3 of 4)

  1. Exercise 1

    Find the roots of the quadratic equations by using the quadratic formula in each of the following:
    (i)
    2x2−3x−5=0\displaystyle 2 x^2-3 x-5=0
    (ii)
    5x2+13x+8=0\displaystyle 5 x^2+13 x+8=0
    (iii)
    −3x2+5x+12=0\displaystyle -3 x^2+5 x+12=0
    (iv)
    −x2+7x−10=0\displaystyle -x^2+7 x-10=0
    (v)
    x2+22x−6=0\displaystyle x^2+2 \sqrt{2} x-6=0
    (vi)
    x2−35x+10=0\displaystyle x^2-3 \sqrt{5} x+10=0
    (vii)
    12x2−11x+1=0\displaystyle \frac{1}{2} x^2-\sqrt{11} x+1=0

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    \(\displaystyle \frac{5}{2},-1\)
    (ii)
    \(\displaystyle -1,-\frac{8}{5}\)
    (iii)
    \(\displaystyle -\frac{4}{3}\), $\displaystyle 3$
    (iv)
    $\displaystyle 5$, $\displaystyle 2$
    (v)
    \(\displaystyle -3 \sqrt{2}, \sqrt{2}\)
    (vi)
    \(\displaystyle \sqrt{5}, 2 \sqrt{5}\)
    (vii)
    \(\displaystyle \sqrt{11}+3, \sqrt{11}-3\)
    \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \](i) \[2x^2-3x-5=0 \] \[x=\frac{3\pm\sqrt{9+40}}{4}=\frac{3\pm7}{4} \] \[x=\frac{5}{2},\ -1 \](ii) \[5x^2+13x+8=0 \] \[x=\frac{-13\pm\sqrt{169-160}}{10}=\frac{-13\pm3}{10} \] \[x=-1,\ -\frac{8}{5} \](iii) \[-3x^2+5x+12=0 \] \[x=\frac{-5\pm\sqrt{25+144}}{-6}=\frac{-5\pm13}{-6} \] \[x=-\frac{4}{3},\ 3 \](iv) \[-x^2+7x-10=0 \] \[x=\frac{-7\pm\sqrt{49-40}}{-2}=\frac{-7\pm3}{-2} \] \[x=2,\ 5 \](v) \[x^2+2\sqrt2\,x-6=0 \] \[x=\frac{-2\sqrt2\pm\sqrt{8+24}}{2}=\frac{-2\sqrt2\pm4\sqrt2}{2} \] \[x=\sqrt2,\ -3\sqrt2 \](vi) \[x^2-3\sqrt5\,x+10=0 \] \[x=\frac{3\sqrt5\pm\sqrt{45-40}}{2}=\frac{3\sqrt5\pm\sqrt5}{2} \] \[x=2\sqrt5,\ \sqrt5 \](vii) \[\frac12x^2-\sqrt{11}\,x+1=0 \] \[x=\frac{\sqrt{11}\pm\sqrt{11-2}}{1}=\sqrt{11}\pm3 \] \[x=\sqrt{11}+3,\ \sqrt{11}-3 \]Answer: (i) \(\displaystyle \frac52,\ -1\) (ii) \(\displaystyle -1,\ -\frac85\) (iii) \(\displaystyle -\frac43,\ 3\) (iv) \(\displaystyle 2,\ 5\) (v) \(\displaystyle \sqrt2,\ -3\sqrt2\) (vi) \(\displaystyle 2\sqrt5,\ \sqrt5\) (vii) \(\displaystyle \sqrt{11}+3,\ \sqrt{11}-3\)

  2. Exercise 2

    Find the roots of the following quadratic equations by the factorisation method:
    (i)
    2x2+53x−2=0\displaystyle 2 x^2+\frac{5}{3} x-2=0
    (ii)
    25x2−x−35=0\displaystyle \frac{2}{5} x^2-x-\frac{3}{5}=0
    (iii)
    32x2−5x−2=0\displaystyle 3 \sqrt{2} x^2-5 x-\sqrt{2}=0
    (iv)
    3x2+55x−10=0\displaystyle 3 x^2+5 \sqrt{5} x-10=0
    (v)
    21x2−2x+121=0\displaystyle 21 x^2-2 x+\frac{1}{21}=0

    Matches the book, not yet reviewed

    This working reaches the answer NCERT prints. It has not yet been read through by hand.

    NCERT’s answer
    (i)
    \(\displaystyle -\frac{3}{2}, \frac{2}{3}\)
    (ii)
    \(\displaystyle -\frac{1}{2}, 3\)
    (iii)
    \(\displaystyle \sqrt{2},-\frac{\sqrt{2}}{6}\)
    (iv)
    \(\displaystyle \frac{\sqrt{5}}{3},-2 \sqrt{5}\)
    (v)
    \(\displaystyle \frac{1}{21}, \frac{1}{21}\)
    (i) \[2x^2+\frac{5}{3}x-2=0\ \Rightarrow\ 6x^2+5x-6=0 \] \[6x^2+9x-4x-6=0\ \Rightarrow\ 3x(2x+3)-2(2x+3)=0 \] \[(2x+3)(3x-2)=0\quad\Rightarrow\quad x=-\frac{3}{2},\ \frac{2}{3} \](ii) \[\frac{2}{5}x^2-x-\frac{3}{5}=0\ \Rightarrow\ 2x^2-5x-3=0 \] \[2x^2-6x+x-3=0\ \Rightarrow\ 2x(x-3)+1(x-3)=0 \] \[(x-3)(2x+1)=0\quad\Rightarrow\quad x=3,\ -\frac{1}{2} \](iii) \[3\sqrt2\,x^2-5x-\sqrt2=0 \] \[3\sqrt2\,x^2-6x+x-\sqrt2=0\ \Rightarrow\ 3\sqrt2\,x(x-\sqrt2)+1(x-\sqrt2)=0 \] \[(x-\sqrt2)(3\sqrt2\,x+1)=0\quad\Rightarrow\quad x=\sqrt2,\ -\frac{1}{3\sqrt2}=-\frac{\sqrt2}{6} \](iv) \[3x^2+5\sqrt5\,x-10=0 \] \[3x^2+6\sqrt5\,x-\sqrt5\,x-10=0\ \Rightarrow\ 3x(x+2\sqrt5)-\sqrt5(x+2\sqrt5)=0 \] \[(x+2\sqrt5)(3x-\sqrt5)=0\quad\Rightarrow\quad x=-2\sqrt5,\ \frac{\sqrt5}{3} \](v) \[21x^2-2x+\frac{1}{21}=0\ \Rightarrow\ 441x^2-42x+1=0 \] \[441x^2-21x-21x+1=0\ \Rightarrow\ 21x(21x-1)-1(21x-1)=0 \] \[(21x-1)^2=0\quad\Rightarrow\quad x=\frac{1}{21},\ \frac{1}{21} \]Answer: (i) \(\displaystyle -\frac32,\ \frac23\) (ii) \(\displaystyle 3,\ -\frac12\) (iii) \(\displaystyle \sqrt2,\ -\frac{\sqrt2}{6}\) (iv) \(\displaystyle -2\sqrt5,\ \frac{\sqrt5}{3}\) (v) \(\displaystyle \frac{1}{21}\) (repeated root)