Question

Find the nature of the roots of the following quadratic equations:
(i) $2x^2-8x+5=0$
(ii) $3x^2-2\sqrt{6}x+2=0$
(iii) $5x^2-4x+1=0$
(iv) $5x\left(x-2\right)+6=0$
(v) $12x^2-4\sqrt{15}x+5=0$
(vi) $x^2-x+2=0$

Answer 1

(i) The given equation is $2x^2-8x+5=0$.

This is of the form $a{x}^2+bx+c=0$, where a = 2, b = −8 and c = 5.

∴ Discriminant, D = $b^2-4ac={\left(-8\right)}^2-4\times 2\times 5=64-40=24>0$

Hence, the given equation has real and unequal roots.

(ii) The given equation is $3x^2-2\sqrt{6}x+2=0$.

This is of the form $a{x}^2+bx+c=0$, where a = 3, b = $-2\sqrt{6}$ and c = 2.

∴ Discriminant, D = $b^2-4ac={\left(-2\sqrt{6}\right)}^2-4\times 3\times 2=24-24=0$

Hence, the given equation has real and equal roots.

(iii) The given equation is $5x^2-4x+1=0$.

This is of the form $a{x}^2+bx+c=0$, where a = 5, b = −4 and c = 1.

∴ Discriminant, D = $b^2-4ac={\left(-4\right)}^2-4\times 5\times 1=16-20=-4<0$

Hence, the given equation has no real roots.

(iv) The given equation is

$$\begin{gathered} 5x\left(x-2\right)+6=0 \\ \Rightarrow 5x^2-10x+6=0 \end{gathered}$$

This is of the form $a{x}^2+bx+c=0$, where a = 5, b = −10 and c = 6.

∴ Discriminant, D = $b^2-4ac={\left(-10\right)}^2-4\times 5\times 6=100-120=-20<0$

Hence, the given equation has no real roots.

(v) The given equation is $12x^2-4\sqrt{15}x+5=0$.

This is of the form $a{x}^2+bx+c=0$, where a = 12, b = $-4\sqrt{15}$ and c = 5.

∴ Discriminant, D = $b^2-4ac={\left(-4\sqrt{15}\right)}^2-4\times 12\times 5=240-240=0$

Hence, the given equation has real and equal roots.

(vi) The given equation is $x^2-x+2=0$.

This is of the form $a{x}^2+bx+c=0$, where a = 1, b = −1 and c = 2.

∴ Discriminant, D = $b^2-4ac={\left(-1\right)}^2-4\times 1\times 2=1-8=-7<0$

Hence, the given equation has no real roots.