9. Show that the following equations have real roots. $$ \begin{array}{ll} ext { a. }(x-a)(x-b)=a b x^{2} & ext { b. } x^{2}-2 a x+a^{2}-b^{2}-c^{2}=0 \ ext { 10. a. Show that the roots of the equation } 2\left(a^{2}+b^{2} ight) x^{2}+2(a+b) x+1=0 ext { can never be real } \ ext { unless } a=b ext {. }\end{array} $$ $$ \begin{array}{l} ext { b. Prove that the roots of the equation } x^{2}+(2 k-1) x+k^{2}=0 ext { are real roots if } k \leq \frac{1}{4} ext {. }\end{array} $$.

Question

9. Show that the following equations have real roots. $$ \begin{array}{ll}	ext { a. }(x-a)(x-b)=a b x^{2} & 	ext { b. } x^{2}-2 a x+a^{2}-b^{2}-c^{2}=0 \ 	ext { 10. a. Show that the roots of the equation } 2\left(a^{2}+b^{2}ight) x^{2}+2(a+b) x+1=0 	ext { can never be real } \ 	ext { unless } a=b 	ext {. }\end{array} $$ $$ \begin{array}{l}	ext { b. Prove that the roots of the equation } x^{2}+(2 k-1) x+k^{2}=0 	ext { are real roots if } k \leq \frac{1}{4} 	ext {. }\end{array} $$.

Cross Wells · Accepted Answer

For Problem 9a, the equation $$(x-a)(x-b) = abx^2$$ has real roots for all real $$a$$ and $$b$$. For Problem 9b, the equation $$x^2 - 2ax + a^2 - b^2 - c^2 = 0$$ has real roots. For Problem 10a, the roots of the equation $$2(a^2 + b^2)x^2 + 2(a+b)x + 1 = 0$$ are not real unless $$a = b$$. For Problem 10b, the roots of the equation $$x^2 + (2k-1)x + k^2 = 0$$ are real if $$k \leq \frac{1}{4}$$.

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