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Neokaal Study

Quadratic Equations

Practice Worksheet

Factorisation, the quadratic formula, discriminants, roots, and application problems.

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Answer all questions.

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Use a separate notebook for your solutions.

  1. Which equation is not quadratic after simplification?

    1. (A)$x^3 + x = (x + 1)^2$
    2. (B)$x(x + 2) = 3x - 2$
    3. (C)$(x - 1)^2 + 2 = 3x$
    4. (D)$(2x + 1)(x - 2) = x + 4$

  2. Solve $2x^2 - 4x - 16 = 0$ using factorisation.

  3. Determine the nature of the roots of $x^2 - 16 = 0$

    1. (A)no real roots
    2. (B)two equal real roots
    3. (C)two distinct real roots
    4. (D)more than two real roots

  4. If a person's age were 6 years less, the square of that age would be 4 more than 6 times the present age. Find the present age.

  5. A train covers 330 km at a constant speed. At 5 km/h faster, it would take 55 minutes less. Find its original speed.

  6. A parent's present age is 2 more than the square of the child's present age. When the child reaches the parent's present age, the parent will be 4 more than 23 times the child's present age. Find both present ages.

  7. Which equation has $-3$ as a root?

    1. (A)$3x^2 - 9x - 54 = 0$
    2. (B)$3x^2 - 9x - 56 = 0$
    3. (C)$3x^2 - 9x - 51 = 0$
    4. (D)$3x^2 - 9x - 53 = 0$

  8. Is $-1$ a root of $x^2 - 1 = 0$? Justify your answer.

  9. If $1$ is a root of $3x^2 + kx - 10 = 0$, find $k$.

    1. (A)$-7$
    2. (B)$7$
    3. (C)$6$
    4. (D)$8$

  10. Which equation has sum of roots equal to $1$?

    1. (A)$3x^2 - 3x - 12 = 0$
    2. (B)$2x^2 - 6x - 4 = 0$
    3. (C)$2x^2 - 4x - 6 = 0$
    4. (D)$x^2 - 5 = 0$

  11. Which equation has two distinct real roots?

    1. (A)$x^2 + 4x - 5 = 0$
    2. (B)$x^2 + 10x + 25 = 0$
    3. (C)$x^2 + 12x + 36 = 0$
    4. (D)$x^2 + 6x + 10 = 0$

  12. Which equation has no real roots?

    1. (A)$3x^2 - 7x + 6 = 0$
    2. (B)$x^2 - 16x + 64 = 0$
    3. (C)$x^2 - 8x + 12 = 0$
    4. (D)$x^2 - 14x + 49 = 0$

  13. For which values of $k$ does $x^2-kx+25=0$ have equal roots?

    1. (A)$k=-10$ or $k=10$
    2. (B)$k=-10$ only
    3. (C)$k=25$
    4. (D)$k=10$ only

  14. Does $x^2 - 4x + 5 = 0$ have two distinct real roots? Justify without solving for the roots.

  15. True or false? Every quadratic equation has at most two distinct real roots. Justify your answer.

  16. For $x^2 + bx + c = 0$, suppose $b=0$ and $c=-8$. Must the two roots be equal in magnitude and opposite in sign? Explain.

  17. When completing the square directly in $x^2 +6x$, which constant must be added and subtracted?

    1. (A)$9$
    2. (B)$3$
    3. (C)$12$
    4. (D)$11$

  18. Solve $2x^2 + 18x + 40 = 0$ using the quadratic formula.

Neokaal Study

Answer Key

  1. Option A: $x^3 + x = (x + 1)^2$

    1. Expand both sides and bring every term to one side.
    2. Its simplified form does not have degree two.
    3. Therefore, option A is the required equation.

  2. The roots are $x = -2$ or $x = 4$.

    1. Clear any fractional coefficients, then split the middle term and factorise.
    2. Set each linear factor equal to zero.
    3. Hence, $x = -2$ or $x = 4$.

  3. The equation has two distinct real roots (option C).

    1. Calculate $D=b^2-4ac=64$.
    2. $D>0$ gives two distinct real roots, $D=0$ gives equal roots, and $D<0$ gives no real roots.
    3. Therefore, this equation has two distinct real roots.

  4. The present age is $16$ years.

    1. Let the present age be $a$. Then $(a-6)^2=6a+4$.
    2. The resulting quadratic has roots $16$ and $2$.
    3. Reject the non-positive value $2$; the present age is $16$ years.

  5. The original speed is $40$ km/h.

    1. Let the original speed be $v$ km/h. The time difference is $330/v-330/(v+5)=\frac{11}{12}$ hour.
    2. Clearing denominators gives a quadratic whose roots are $40$ and $-45$.
    3. A speed must be positive, so the original speed is $40$ km/h.

  6. The child is $12$ years old and the parent is $146$ years old.

    1. Let the child's age be $c$; then the parent's age is $c^2+2$.
    2. When the child reaches that age, the parent's age will be $2(c^2+2)-c$. Use the stated comparison to form the quadratic.
    3. The valid age is $c=12$, giving parent age $12^2+2=146$ years.

  7. Option A: $3x^2 - 9x - 54 = 0$

    1. Substitute $x = -3$ into each equation.
    2. For option A, the left side becomes $0$.
    3. The other options give non-zero values, so they do not have the stated root.

  8. Yes.

    1. Substitute $x = -1$ into the left side.
    2. The resulting value is $0$.
    3. Since the value is $0$, the number is a root.

  9. $k = 7$ (option B).

    1. Substitute $x = 1$ in the equation.
    2. This gives $3 + 1k + -10 = 0$.
    3. Solving the linear equation gives $k = 7$.

  10. Option A: $3x^2 - 3x - 12 = 0$

    1. For $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$.
    2. For option A, $-b/a = 1$.
    3. The coefficient ratios in the other options give different sums.

  11. Option A: $x^2 + 4x - 5 = 0$

    1. Compute $D=b^2-4ac$ for each option.
    2. For option A, $D=36$.
    3. This discriminant corresponds to two distinct real roots.

  12. Option A: $3x^2 - 7x + 6 = 0$

    1. Compute $D=b^2-4ac$ for each option.
    2. For option A, $D=-23$.
    3. This discriminant corresponds to no real roots.

  13. $k=-10$ or $k=10$ (option A).

    1. Equal roots require the discriminant to be zero.
    2. Here, $D=(-k)^2-4(25)=k^2-100$.
    3. Solving $D=0$ gives $k=-10$ or $k=10$.

  14. No.

    1. Its discriminant is $D=b^2-4ac=-4$.
    2. Two distinct real roots require $D>0$.
    3. Here $D$ is negative, so the answer is no.

  15. True.

    1. Use the discriminant or a counterexample to test the statement.
    2. A non-zero polynomial of degree two cannot have more than two distinct roots.
    3. Therefore, the statement is true.

  16. Yes. The roots are equal in magnitude and opposite in sign.

    1. The equation becomes $x^2 - 8=0$.
    2. Thus $x^2=8$, so $x=\sqrt{8}$ or $x=-\sqrt{8}$.
    3. The two values have the same magnitude and opposite signs.

  17. Add and subtract $9$ (option A).

    1. Write $1x^2 +6x$ as the beginning of $(1x + m)^2$.
    2. Matching the middle term gives $m=3$.
    3. Therefore the required constant is $m^2=9$.

  18. The roots are $x = -5$ or $x = -4$.

    1. Use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ with $a=2$, $b=18$, and $c=40$.
    2. The discriminant is $D=4$.
    3. Hence, $x = -5$ or $x = -4$.