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(B) Given the quadratic equation: $a x^2+(a+b) x+b = 0$Factoring the expression:$a x^2 + a x + b x + b = 0$$a x(x + 1) + b(x + 1) = 0$$(a x + b)(x + 1

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$I$ and $III$ only

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(B) Given the quadratic equation: $a x^2+(a+b) x+b = 0$Factoring the expression:$a x^2 + a x + b x + b = 0$$a x(x + 1) + b(x + 1) = 0$$(a x + b)(x + 1) = 0$The roots are $x = -\frac{b}{a}$ and $x = -1$.Analysis:$1$. Since $-1$ is a root,the equation always has at least one negative root. Thus,statement $I$ is true.$2$. The roots are $-1$ and $-\frac{b}{a}$. Both are real numbers because $a$ and $b$ are real. Thus,statement $III$ is true.$3$. The existence of a positive root depends on the sign of $-\frac{b}{a}$. If $\frac{b}{a} > 0$,the root is negative. If $\frac{b}{a} Therefore,statements $I$ and $III$ are necessarily true.

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.

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Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?$I$. It has at least one negative root.$II$. It has at least one positive root.$III$. Both its roots are real.