Quadratics – Practice questions

Q1. If the roots of a quadratic equation  2x2+3xk=0 are tanα  and  tanβ , find the value of the expression  tanα+β  in terms of  k and hence calculate the value of k if  α+β=3π4

Q2. Find the value of x in the form of   a+bc where  a,b,c if


Q3. Find the value of  x in the form of a+b where  a, b if 


Q4. A quadratic equation is given by  bcx2+cax+ab=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 x6/59x3/5+8=0 , where x .

Q6. If   a>0, b>0  and  c>0 with  a, b, c for the quadratic equation ax2+bx+c=0 and both roots are real, show that both roots have to be negative.

Q7. The equations x2+x+k=0  and  kx22kx+1=0 have a common root α . Find the value of α  in terms of  k .

Q8. Find the values  k  of for which  x2+x+k0 .

Q9. Let the roots of  lx2+mx+n=0 be  α, β  and roots of  ax2+bx+c=0 be γ, δ . If  α, β, γ, δ  are in arithmetic sequence, show that b2 4aca2=m2 4nll2 .

Q10. If   x+ky=1ax2+by2=1 has only one solution, show that b=ak2a  1

Q11. If x2+2xk=0 has roots  sinα, cosα then find the value of k  .

Q12. Show that there is no value of  a for which xa2+1=0 can have repeated roots. 

Q13. If α, β are the roots of the equation , show that the equation with roots α+k, β+k is x2kx1=0 .

Q14. Write the following quadratic function  fx=6x2+5x6  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  ax2+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 ba  and product of the roots is  ca

(c) find the solution of  ax2+bx+c0  when

(i)  a<0

(ii)  a>0.