If the discriminant of ax^2+bx+c=0 is negative, which statement is true about its roots?

Both roots are real and distinct

One root is real and the other is complex

Quadratic Equations

Practise Quadratic Equations for FBISE Grade 9–10 Mathematics (Grade 9–10): 5 multiple-choice questions with instant checking and worked explanations. Free, no sign-up.

FBISE Grade 9–10MathematicsGrade 9–105 questions

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Question 1

The quadratic equation $x^{2} + m x + n = 0$ has two distinct integer roots whose difference is $4$ . The quadratic $2 x^{2} + (m + 2) x + (n - 3) = 0$ has a repeated root. If $m$ is negative, what is the value of $m$ ?
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Answer: D) -6
Using Vieta’s formulas, the sum of the integer roots equals $- m$ and their product equals $n$ . Because the roots differ by 4, write them as $r$ and $r + 4$ , giving $m = - (2 r + 4)$ and $n = r (r + 4)$ . The second quadratic has a repeated root, so its discriminant is zero: $(m + 2)^{2} = 8 (n - 3)$ . Substituting $m$ and $n$ leads to the quadratic $r^{2} + 6 r - 7 = 0$ , whose integer solutions are $r = 1$ and $r = - 7$ . Only $r = 1$ yields a negative $m$ , namely $m = - 6$ , which is the correct answer. The other options arise from common algebraic sign or discriminant mistakes.
Question 2

If the constant term of a quadratic function $f (x) = a x^{2} + b x + c$ is increased, which of the following must change?
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Answer: C) The y-intercept
The constant term $c$ is exactly the y‑intercept of the parabola $y = a x^{2} + b x + c$ , because setting $x = 0$ gives $y = c$ . Raising $c$ moves the whole graph upward, changing the point where it crosses the y‑axis. The axis of symmetry depends only on $a$ and $b$ , the opening direction on the sign of $a$ , and the x‑intercepts on the discriminant, none of which are forced to change when $c$ changes.
Question 3

If the discriminant of $a x^{2} + b x + c = 0$ is negative, which statement is true about its roots?
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Answer: C) Roots are complex conjugates
A negative discriminant $Δ = b^{2} - 4 a c < 0$ means the expression under the square root in the quadratic formula is negative, producing non‑real numbers. The two resulting roots are complex and occur as a conjugate pair, so the statement "Roots are complex conjugates" is true.
Question 4

The quadratic $3 x^{2} - 12 x + k = 0$ has one root twice the other. Find $k$ .
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Answer: B) 32/3
Dividing the equation by 3 gives a monic quadratic. Using Vieta, the sum of the roots equals $4$ , so with roots $a$ and $2 a$ we find $a = 4/3$ . The product equals $k /3$ , and $a \cdot 2 a = 32/9$ , giving $k = 32/3$ . Thus option $32/3$ is correct.
Question 5

If $x = 2$ is a root of $x^{2} + p x + q = 0$ and the other root equals three times the sum of the coefficients, what is $p + q$ ?
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Answer: D) -0.5
Using Vieta’s formulas, the sum of the roots equals $- p$ and the product equals $q$ . Substituting the given relation for the second root and solving the resulting linear system yields $p = - 7/2$ and $q = 3$ , so $p + q = - 1/2$ . Therefore the correct choice is –0.5.

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