Exercises 4 (Quadratic equations)

  1. Draw the graphs of the quadratic functions given below and indicate clearly where the curves intersect the \(x\) and \(y\) axes.

    1. \(y=(x+1)^2-1\)
    2. \(y=-(x+1)^2-1\)
    3. \(y=(x+2)^2-3\)
    4. \(y=(x+2)^2+3\)
    5. \(y=-(x+2)^2+3\)

  2. Solve the following quadratic equations by the method of factorisation:

    1. \(x^2-x-6=0\)
    2. \(x^2-16=0\)
    3. \(x^2-2x=0\)
    4. \(x^2-6x+9=0\)
    5. \(6x^2+18x+12=0\)
    6. \(6p^2-31p+35=0\)
    7. \(6x^2-11x-7=0\)
    8. \(-3r^2-14r+5=0\)
    9. \(14x^2=29x-12\)

  3. Solve the following quadratic equations using the formula:

    1. \(3x^2-8x+2=0\)
    2. \(-2x^2+3x+7=0\)
    3. \(4x^2-3x-2=0\)
    4. \(7r^2+8r-2=0\)
    5. \(x^2+x+\frac14=\frac19\)
    6. \(5x^2-4x-1=0\)
    7. \(2a^2-5.3a+1.25=0\)
    8. \(x(x+4)+2x(x+3)=5\)
    9. \(\frac{3}{2x-3}-\frac{2}{x+1}=5\)
    10. \(\frac{2}{x+2}+\frac{3}{x+1}=5\)

  4. Solve the following non-linear simultaneous equations:

    1. \(p-2q=1\),    \(p^2-3pq+4q^2=11\)
    2. \(x+y=1\),    \(3x^2-xy+y^2=37\)
    3. \(x-y=2\),    \(x^3-y^3=152\)
    4. \(y=x^2+5x-3\),    \(y=3x-2\)
    5. \(2a^2+ab-b^2=8\),    \(3a+2b=5\)

Problems leading to quadratic equations

  1. The angle in radians turned through by a shaft in \(t\) seconds is given by \[\theta=\omega t+\frac{\alpha t^2}{2}.\] Determine the time taken to complete five revolutions given that \(\omega=2.7\, \mathrm{rad/s}\) and \(\alpha=0.8\, \mathrm{rad/s^2}\).

  2. In a right angled triangle the hypotenuse is twice as long as one of the sides forming the right angle. The remaining side is \(80\,\mathrm{mm}\) long. Calculate the area of the triangle.

  3. The shape shown below has an area of \(600\,\mathrm{mm^2}\). Determine the radius \(r\).

  1. The total surface area of a cylinder whose ends are enclosed is \(0.29\,\mathrm{m^2}\). If the height of the cylinder is \(75\,\mathrm{mm}\), determine the radius.

  2. If the area of the shape below is \(9693\,\mathrm{mm^2}\), determine the radius \(r\).

  1. A motorist travels \(84\,\mathrm{km}\) from one city to another. On the return journey, the average speed was increased by \(4\,\mathrm{km/h}\) and the journey took \(30\) minutes less. What was the average speed for the first part of the trip and how long did it take for the double journey?

Solutions: 2. (i) \(x=-2,x=3\); (ii) \(x=4,x=-4\); (iii) \(x=0,x=2\); (iv) \(x=3\) (double); (v) \(x=-1,x=-2\); (vi) \(p=7/2, p=5/3\); (vii) \(x=7/3,x=-1/2\); (viii) \(r=1/3,r=-5\); (ix) \(x=4/7,x=3/2\)

3. (i) \(x=2.39,x=0.28\); (ii) \(x=-1.26,x=2.76\); (iii) \(x=1.17,x=-0.42\); (iv) \(x=0.21,x=-1.35\); (v) \(x=-1/6,x=-5/6\); (vi) \(x=1,x=-1/5\); (vii) \(a=2.39,a=0.26\); (viii) \(x=0.44,x=-3.77\); (ix) \(x=1.76,x=-1.36\); (x) \(x=-0.22,x=-1.77\)

4. (i) \(p=-4,q=-5/2\) or \(p=5,q=2\); (ii) \(x=3,y=-2\) or \(x=-12/5,y=17/5\); (iii) \(x=6,y=4\) or \(x=-4,y=-6\); (iv) \(x=0.41,y=-0.77\) or \(x=-2.41,y=-9.23\); (v) \(a=3,b=-2\) or \(a=2.7,b=-1.55\)

5. \(t=6.1\,\mathrm{s}\); 6. \(1848\,\mathrm{mm}^2\); 7. \(r=4.36\,\mathrm{mm}\); 8. \(r=180\,\mathrm{mm}\); 9. \(r=71.86\,\mathrm{mm}\) or \(r=30\,\mathrm{mm}\); 10. \(6.5\) hours