Quadratics – Practice questions

Q1. If the roots of a quadratic equation  $2x^2+3x–k=0$ are $\tan \alpha$  and  $\tan \beta$ , find the value of the expression  $\tan \left(\alpha +\beta \right)$  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  $a,b,c\in \mathrm{\mathbb{Z}}$ 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  $a, b\in \mathrm{\mathbb{Z}}$ if 

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

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

Q6. If   $a>0, b>0$  and  $c>0$ with  $a, b, c\in \mathrm{ℝ}$ 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  $\alpha , \beta$  and roots of  $a{x}^2+bx+c=0$ be $\gamma , \delta$ . If  $\alpha , \beta , \gamma , \delta$  are in arithmetic sequence, show that $\frac{b^2– 4ac}{a^2}=\frac{m^2– 4nl}{l^2}$ .

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

Q11. If $x^2+2x–k=0$ has roots  $\sin \alpha , \cos \alpha$ 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 $\alpha , \beta$ are the roots of the equation , show that the equation with roots $\alpha +k, \beta +k$ is $x^2–kx–1=0$ .

Q14. Write the following quadratic function  $f\left(x\right)=6x^2+5x–6$  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  $a{x}^2+bx+c=0, a$ 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)  $a>0.$