(a) Discuss the nature of the roots of the following quadratic equations, without solving the equations: (1) $$ x^{2}-3 x-10=0 $$ (2) $$ x^{2}-2 x+1=0 $$ (3) $$ x^{2}-5 x-7=0 $$ (4) $$ x^{2}-x+3=0 $$ (5) $$ 2 x^{2}+6=-7 x $$ (6) $$ x+\frac{6}{x}=4 $$ (77) $$ (x+2)(x-1)=4 x $$ (8) $$ x^{2}+p^{2}=2 p x $$ (9)) $$ k x^{2}+(4 k-1) x=2-4 k $$ (b) For which value(s) of $$ m $$ will the following expressions have real roots? (1) $$ x^{2}+6 x+m=0 $$ (2) $$ 3 x^{2}=2 m x-3 $$ (3) $$ m x^{2}+m x+m=3 $$ (c) For which value(s) of $$ m $$ will the following expressions have equal roots? (11) $$ x^{2}+2 m=8 x $$ (2) $$ x^{2}+m x+9=0 $$ ((3)) $$ x^{2}-m x+2 x+1=0 $$ (d) For which value(s) of $$ m $$ will the following expressions have non-real roots? (1)) $$ x^{2}-6 x=m $$ (2) $$ x^{2}+m x+4=3 x $$ (3) $$ \frac{x^{2}-5 x+5}{2 x^{2}-x+1}=\frac{1}{m} $$ Show that the equation $$ x^{2}+(k-2) x-k=0 $$ has real roots for all real values of $$ k $$.

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To analyze the nature of the roots of the given quadratic equations, we will use the discriminant $$ D $$ of a quadratic equation of the form $$ ax^2 + bx + c = 0 $$. The discriminant is given by: $$ D = b^2 - 4ac $$ The nature of the roots can be determined as follows: - If $$ D > 0 $$, the equation has two distinct real roots. - If $$ D = 0 $$, the equation has exactly one real root (a repeated root). - If $$ D < 0 $$, the equation has no real roots (the roots are complex). ### Part (a): Nature of the Roots 1. **For $$ x^{2}-3x-10=0 $$**: - Here, $$ a = 1 $$, $$ b = -3 $$, $$ c = -10 $$. - Calculate $$ D = (-3)^2 - 4(1)(-10) = 9 + 40 = 49 $$. - Since $$ D > 0 $$, there are two distinct real roots. 2. **For $$ x^{2}-2x+1=0 $$**: - Here, $$ a = 1 $$, $$ b = -2 $$, $$ c = 1 $$. - Calculate $$ D = (-2)^2 - 4(1)(1) = 4 - 4 = 0 $$. - Since $$ D = 0 $$, there is exactly one real root. 3. **For $$ x^{2}-5x-7=0 $$**: - Here, $$ a = 1 $$, $$ b = -5 $$, $$ c = -7 $$. - Calculate $$ D = (-5)^2 - 4(1)(-7) = 25 + 28 = 53 $$. - Since $$ D > 0 $$, there are two distinct real roots. 4. **For $$ x^{2}-x+3=0 $$**: - Here, $$ a = 1 $$, $$ b = -1 $$, $$ c = 3 $$. - Calculate $$ D = (-1)^2 - 4(1)(3) = 1 - 12 = -11 $$. - Since $$ D < 0 $$, there are no real roots. 5. **For $$ 2x^{2}+6=-7x $$**: - Rearranging gives $$ 2x^{2} + 7x + 6 = 0 $$. - Here, $$ a = 2 $$, $$ b = 7 $$, $$ c = 6 $$. - Calculate $$ D = (7)^2 - 4(2)(6) = 49 - 48 = 1 $$. - Since $$ D > 0 $$, there are two distinct real roots. 6. **For $$ x+\frac{6}{x}=4 $$**: - Multiplying through by $$ x $$ gives $$ x^2 - 4x + 6 = 0 $$. - Here, $$ a = 1 $$, $$ b = -4 $$, $$ c = 6 $$. - Calculate $$ D = (-4)^2 - 4(1)(6) = 16 - 24 = -8 $$. - Since $$ D < 0 $$, there are no real roots. 7. **For $$ (x+2)(x-1)=4x $$**: - Expanding gives $$ x^2 + x - 2 = 4x $$ or $$ x^2 - 3x - 2 = 0 $$. - Here, $$ a = 1 $$, $$ b = -3 $$, $$ c = -2 $$. - Calculate $$ D = (-3)^2 - 4(1)(-2) = 9 + 8 = 17 $$. - Since $$ D > 0 $$, there are two distinct real roots. 8. **For $$ x^{2}+p^{2}=2px $$**: - Rearranging gives $$ x^2 - 2px + p^2 = 0 $$. - Here, $$ a = 1 $$, $$ b = -2p $$, $$ c = p^2 $$. - Calculate $$ D = (-2p)^2 - 4(1)(p^2) = 4p^2 - 4p^2 = 0 $$. - Since $$ D = 0 $$, there is exactly one real root. 9. **For $$ kx^{2}+(4k-1)x=2-4k $$**: - Rearranging gives $$ kx^{2} + (4k-1)x + (4k-2) = 0 $$. - Here, $$ a = k $$, $$ b = 4k - 1 $$, $$ c = 4k - 2 $$. - Calculate $$ D = (4k - 1)^2 - 4k(4k - 2) = 16k^2 - 8k + 1 - (16k^2 - 8k) = 1 $$. - Since $$ D > 0 $$, there are two distinct real roots. ### Part (b): Values of $$ m $$ for Real Roots 1. **For $$ x^{2}+6x+m=0 $$**: - $$ D = 6^2 - 4(1)(m) = 36 - 4m \geq 0 $$. - Thus, $$ 36 \geq 4m $$ or $$ m \leq 9 $$. 2. **For $$ 3x^{2}=2mx-3 $$**: - Rearranging gives $$ 3x^{2} - 2mx + 3 = 0 $$. - $$ D = (-2m)^2 - 4(3)(3) = 4m^2 - 36 \geq 0 $$. - Thus, $$ 4m^2 \geq 36 $$ or $$ m^2 \geq 9 $$ leading to $$ m \leq -3 $$ or $$ m \geq 3 $$. 3. **For $$ mx^{2}+mx+m=3 $$**: - Rearranging gives $$ mx^{2} + mx + (m - 3) = 0 $$. - $$ D = m^2 - 4m(m - 3) = m^2 - 4m^2 + 12m = -3m^2 + 12m \geq 0 $$. - Thus, $$ -3m(m - 4) \geq 0 $$ leading to $$ m \leq 0 $$ or $$ m \geq 4 $$. ### Part (c): Values of $$ m $$ for Equal Roots 1. **Part (a): Nature of the Roots** 1. **$$ x^{2}-3x-10=0 $$**: Two distinct real roots. 2. **$$ x^{2}-2x+1=0 $$**: One real root. 3. **$$ x^{2}-5x-7=0 $$**: Two distinct real roots. 4. **$$ x^{2}-x+3=0 $$**: No real roots. 5. **$$ 2x^{2}+6=-7x $$**: Two distinct real roots. 6. **$$ x+\frac{6}{x}=4 $$**: No real roots. 7. **$$ (x+2)(x-1)=4x $$**: Two distinct real roots. 8. **$$ x^{2}+p^{2}=2px $$**: One real root. 9. **$$ kx^{2}+(4k-1)x=2-4k $$**: Two distinct real roots. **Part (b): Values of $$ m $$ for Real Roots** 1. **$$ x^{2}+6x+m=0 $$**: $$ m \leq 9 $$. 2. **$$ 3x^{2}=2mx-3 $$**: $$ m \leq -3 $$ or $$ m \geq 3 $$. 3. **$$ mx^{2}+mx+m=3 $$**: $$ m \leq 0 $$ or $$ m \geq 4 $$. **Part (c): Values of $$ m $$ for Equal Roots** 1. **$$ x^{2}+2m=8x $$**: $$ m = 0 $$ or $$ m = 4 $$. 2. **$$ x^{2}+mx+9=0 $$**: $$ m = \pm 6 $$. 3. **$$ x^{2}-mx+2x+1=0 $$**: $$ m = 3 $$ or $$ m = -1 $$. **Part (d): Values of $$ m $$ for Non-Real Roots** 1. **$$ x^{2}-6x=m $$**: $$ m > 9 $$. 2. **$$ x^{2}+mx+4=3x $$**: $$ m < -6 $$ or $$ m > 6 $$. 3. **$$ \frac{x^{2}-5x+5}{2x^{2}-x+1}=\frac{1}{m} $$**: $$ m \neq 0 $$. **Additional Information:** - The equation $$ x^{2}+(k-2)x-k=0 $$ has real roots for all real values of $$ k $$ because the discriminant $$ D = (k-2)^2 - 4(1)(-k) = k^2 - 4k + 4 + 4k = k^2 + 4 $$ is always positive.

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