# Quadratics – Practice questions

Q1. If the roots of a quadratic equation  $2{x}^{2}+3x–k=0$ are $\mathrm{tan}\alpha$  and  $\mathrm{tan}\beta$ , find the value of the expression  $\mathrm{tan}\left(\alpha +\beta \right)$  in terms of  $k$ and hence calculate the value of $k$ if  $\alpha +\beta =\frac{3\pi }{4}$

Q2. Find the value of $x$ in the form of   $\frac{a+\sqrt{b}}{c}$ where  $a,b,c\in \mathrm{ℤ}$ if

$x=\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{\frac{1}{2}+..........}}}}$

Q3. Find the value of  $x$ in the form of $a+\sqrt{b}$ where  if

$x=\frac{1}{2+\frac{1}{2+\frac{1}{2+........}}}$

Q4. A quadratic equation is given by  $\left(b–c\right){x}^{2}+\left(c–a\right)x+a–b=0,$

(a) Show that  $x=1$ is a root of the equation

(b) Hence, find the other root of the equation.

Q5. Solve the equation ${x}^{6/5}–9{x}^{3/5}+8=0$ , where $x\in \mathrm{ℤ}$ .

Q6. If    and  $c>0$ with  for the quadratic equation $a{x}^{2}+bx+c=0$ and both roots are real, show that both roots have to be negative.

Q7. The equations ${x}^{2}+x+k=0$  and  $k{x}^{2}–2kx+1=0$ have a common root $\alpha$ . Find the value of $\alpha$  in terms of  $k$ .

Q8. Find the values  $k$  of for which  ${x}^{2}+x+k\ge 0$ .

Q9. Let the roots of  $l{x}^{2}+mx+n=0$ be   and roots of  $a{x}^{2}+bx+c=0$ be . If   are in arithmetic sequence, show that .

Q10. If   $x+ky=1$$a{x}^{2}+b{y}^{2}=1$ has only one solution, show that

Q11. If ${x}^{2}+2x–k=0$ has roots  then find the value of $k$  .

Q12. Show that there is no value of  $a$ for which ${\left(x–a\right)}^{2}+1=0$ can have repeated roots.

Q13. If are the roots of the equation , show that the equation with roots is ${x}^{2}–kx–1=0$ .

Q14. Write the following quadratic function  $f\left(x\right)=6{x}^{2}+5x–6$  in:

(a) intercept form

(b) vertex form/turning point form

(c) find the zeroes and the vertex of the above quadratic.

Q15. Using the method of completing squares, solve the equation  is not equal to  $0$

(a) and write the conditions for the above equation to have:

(i) real and distinct roots

(ii) real and equal roots

(iii) no real roots

(b) show that the sum of the roots is given by $–\frac{b}{a}$  and product of the roots is  $\frac{c}{a}$

(c) find the solution of  $a{x}^{2}+bx+c\le 0$  when

(i)  $a<0$

(ii)  $a>0.$