taiwan-gsat 2025 Q9

taiwan-gsat · Other · gsat__math-a 5 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count)

Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options.
(1) $c < 0$
(2) $b < 0$
(3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots
(4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots
(5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots

Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options.\\
(1) $c < 0$\\
(2) $b < 0$\\
(3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots\\
(4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots\\
(5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots

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