Quadratic Equations: practice questions

6 questions · 11 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   If the equation $x^2 - 6x + k = 0$ has equal roots, find $k$.

   A

   $3$

   B

   $6$

   C

   $9$

   D

   $36$

2. Question 2Multiple choice · 1 mark

   $\alpha$ and $\beta$ are the roots of $2x^2 - 5x - 3 = 0$. Find $\alpha^2 + \beta^2$.

   A

   $\dfrac{19}{4}$

   B

   $\dfrac{37}{4}$

   C

   $\dfrac{13}{4}$

   D

   $\dfrac{49}{4}$

3. Question 3Multiple choice · 1 mark

   Which of the following equations has no real roots?

   A

   $x^2 + 4x + 4 = 0$

   B

   $x^2 - 3x - 1 = 0$

   C

   $x^2 - 5 = 0$

   D

   $2x^2 + x + 1 = 0$

4. Question 4Multiple choice · 1 mark

   Solve $x(x - 3) = 2(x - 3)$.

   A

   $x = 2$ or $x = 3$

   B

   $x = 2$ only

   C

   $x = 3$ only

   D

   $x = -2$ or $x = 3$

5. Question 53 marks

   The equation $x^2 + (k - 2)x + 1 = 0$ has two distinct real roots. Find the range of values of $k$.

   Show worked solution

   Two distinct real roots means $\Delta > 0$:

   $$(k - 2)^2 - 4(1)(1) > 0 \qquad \text{1M}$$ $$k - 2 > 2 \text{ or } k - 2 < -2 \qquad \text{1M}$$ $$k > 4 \text{ or } k < 0 \qquad \text{1A}$$
6. Question 64 marks

   $\alpha$ and $\beta$ are the roots of $x^2 - 4x + 1 = 0$. Form a quadratic equation in $x$ whose roots are $\alpha^2$ and $\beta^2$.

   Show worked solution

   $\alpha + \beta = 4$ and $\alpha\beta = 1$. 1M

   $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 16 - 2 = 14$ 1A

   $\alpha^2\beta^2 = (\alpha\beta)^2 = 1$ 1A

   The required equation is

   $$x^2 - 14x + 1 = 0 \qquad \text{1A}$$