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If the difference between the roots of $a x^{2}+b x+c=0$ is 1 , then which one of the following is correct?
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Verified Answer
The correct answer is:
$\quad b^{2}=a(a+4 c)$
Let $\alpha$ and $\beta$ be the roots of $a x^{2}+b x+c=0$ $\alpha+\beta=-\mathrm{b} / \mathrm{a}$
$\alpha \beta=\mathrm{c} / \mathrm{a}$
Given $\alpha-\beta=1$
Consider $(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$
$\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}-1=\frac{4 \mathrm{c}}{\mathrm{a}}$
$b^{2}-a^{2}=4 a c$
$b^{2}=a(a+4 c)$
$\alpha \beta=\mathrm{c} / \mathrm{a}$
Given $\alpha-\beta=1$
Consider $(\alpha+\beta)^{2}-(\alpha-\beta)^{2}=4 \alpha \beta$
$\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}-1=\frac{4 \mathrm{c}}{\mathrm{a}}$
$b^{2}-a^{2}=4 a c$
$b^{2}=a(a+4 c)$
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