Quadratic function

Quadratic equations

1. Solve the following quadratic (and higher order) equations:
   

   (a)   $(x-2)(x-5)=0$

   (b)   $(x-4)(x+1)=0$

   (c)   $(x-1)(x-4)(x+6)=0$

   (d)   $x(x-3)(x+7)=0$

   Solutions:    (a)  $x_1=2,x_2=5$;     (b)  $x_1=4,x_2=-1$;     (c)  $x_1=1,x_2=4,x_3=-6$;     (d)  $x_1=0,x_2=3,x_3=-7$
2. Solve the following equations using factorisation:
   

   (a)   $x^2-5x+6=0$

   (b)   $x^2+2x-15=0$

   (c)   $x^3-2x^2-8x=0$

   Solutions:    (a)  $x_1=2,x_2=3$;     (b)  $x_1=3,x_2=-5$;     (c)  $x_1=0,x_2=-2,x_3=4$
3. Solve the following equations:
   

   (a)   $x^2=x+2$

   (b)   $x^2=6x$

   (c)   $x^3=16x$

   (d)   $x^3+45=5x^2+9x$

   Solutions:    (a)  $x_1=2,x_2=-1$;     (b)  $x_1=0,x_2=6$;     (c)  $x_1=0,x_2=4,x_3=-4$;     (d)  $x_1=5,x_2=3,x_3=-3$
4. Solve the following equations:
   

   (a)   $2x^2=10x-8$

   (b)   $2x^2=3x-1$

   (c)   $3x^2+6=11x$

   Solutions:    (a)  $x_1=1,x_2=4$;     (b)  $x_1=\frac{1}{2},x_2=1$;     (c)  $x_1=\frac{2}{3},x_2=3$
5. Solve the following equations by completing the square:
   

   (a)   $x^2-4x+3=0$

   (b)   $x^2-6x+5=0$

   (c)   $x^2-6x+4=0$

   (d)   $x^2-8x+8=0$

   Solutions:    (a)  $x_1=1,x_2=3$;     (b)  $x_1=1,x_2=5$;     (c)  $x_1=3-\sqrt{5},x_2=3+\sqrt{5}$;     (d)  $x_1=4-2\sqrt{2},x_2=4+2\sqrt{2}$
6. Solve the following equations using the quadratic formula:
   

   (a)   $3x^2+x-2=0$

   (b)   $8x^2-14x+3=0$

   (c)   $2x^2-8x+2=0$

   (d)   $3x^2-9x+3=0$

   Solutions:    (a)  $x_1=-1,x_2=\frac{2}{3}$;     (b)  $x_1=\frac{3}{2},x_2=\frac{1}{4}$;     (c)  $x_1=2-\sqrt{3},x_2=2+\sqrt{3}$;     (d)  $x_1=\frac{3-\sqrt{5}}{2},x_2=\frac{3+\sqrt{5}}{2}$
7. Calculate the discriminant and find out how many real roots do the following equations have:
   

   (a)   $5x^2-x+2=0$

   (b)   $4x^2+8x-5=0$

   (c)   $9x^2-12x+4=0$

   Solutions:    (a)  $\mathrm{\Delta }=-39$, no real roots;     (b)  $\mathrm{\Delta }=144$, two real roots;     (c)  $\mathrm{\Delta }=0$, one repeated root
8. Calculate the discriminant and solve the following equations:
   

   (a)   $x^2-8x+13=0$

   (b)   $x^2-36x+324=0$

   (c)   $6x^2-19x+15=0$

   Solutions:    (a)  $\mathrm{\Delta }=12,x_1=4+\sqrt{3},x_2=4-\sqrt{3}$;     (b)  $\mathrm{\Delta }=0,x_1=x_2=18$;     (c)  $\mathrm{\Delta }=1,x_1=\frac{3}{2},x_2=\frac{5}{3}$
9. Find the value of the constant $m$ given that the following equation has one repeated root:
   

   (a)   $x^2-10x+(3m+4)=0$

   (b)   $4x^2+mx+m+5=0$

   Solutions:    (a)  $m=7$;     (b)  $m_1=-4,m_2=20$
10. Solve the following equations. Write your answers in exact form:
    

    (a)   $(x-1)(x-3)=6$

    (b)   $(x+2{)}^3=x^3-5x-4$

    Solutions:    (a)  $x_1=2-\sqrt{7},x_2=2+\sqrt{7}$;     (b)  $x_1=-\frac{3}{2},x_2=-\frac{4}{3}$
11. Solve the following equations. Write your answers correct to three significant figures:
    

    (a)   $2x(x-7)+15=0$

    (b)   $(2x+1)(x-2)=(x+1{)}^2$

    Solutions:    (a)  $x_1=5,68,x_2=1,32$;     (b)  $x_1=5,54,x_2=-0,541$
12. Solve the following equations:
    

    (a)   $5x-4=\frac{1}{x}$

    (b)   $x-2=\frac{20}{x-1}$

    (c)   ${\displaystyle \frac{x}{x-2}=\frac{4}{x-5}}$

    Solutions:    (a)  $x_1=-\frac{1}{5},x_2=1$;     (b)  $x_1=-3,x_2=6$;     (c)  $x_1=1,x_2=8$
13. Solve the following equations:
    

    (a)   $2x^4-5x^2-12=0$

    (b)   $(x^2+1{)}^2-12(x^2+1)+20=0$

    (c)   $x-7\sqrt{x}+10=0$

    Solutions:    (a)  $x_1=2,x_2=-2$;     (b)  $x_1=1,x_2=-1,x_3=3,x_4=-3$;     (c)  $x_1=4,x_2=25$
14. The product of two positive numbers is 120. The first number is 7 more than the other. Find these two numbers.
    Solutions:    $a=15,b=8$
15. The product of two positive numbers is 403. The first number is 5 more than twice the other number. Find these two numbers.
    Solutions:    $a=31,b=13$
16. The product of two numbers is 360. The first number is 3 more than one half of the other number. Find these two numbers.
    Solutions:    $a_1=15,b_1=24$;     $a_2=-12,b_2=-30$
17. The area of a rectangle is $2100{\mathrm{cm}}^2$. If the side $a$ were $10\mathrm{cm}$ longer it would be twice as long as the side $b$. Find the lengths of sides $a$ and $b$.
    Solutions:    $a=60\mathrm{cm},b=35\mathrm{cm}$
18. A farmer has two fields. Each of them has the form of a square. The side of the first field is $20\mathrm{m}$ longer than the other. Both fields together have the area of $51400{\mathrm{m}}^2$. Determine the area of the first field and the area of the second field.
    Solutions:    $a^2=28900{\mathrm{m}}^2,b^2=22500{\mathrm{m}}^2$

Graphs of quadratic functions

19. Draw graphs of the following functions:
    

    (a)   $f(x)=x^2-4$

    (b)   $f(x)=x^2-4x$

    (c)   $f(x)=x^2-4x+3$

20. Draw graphs of the following functions:
    

    (a)   $y=x^2-2x-3$

    (b)   $y=-x^2+6x-8$

    (c)   $y=\frac{1}{2}{x}^2-2x+2$

21. Draw graphs of the following functions:
    

    (a)   $y=2x^2-4x-1$

    (b)   $y=x^2-4x+5$

    (c)   $y=2x^2-3x+2$

22. Quadratic function $f$ has the vertex $V(2,-4)$ and passes through the point $P(6,4)$.
    

    (a)   Write the equation of this quadratic function in the form $f(x)=a{x}^2+bx+c$.

    (b)   Draw the graph of $f$.

    Solutions:    (a)  $f(x)=\frac{1}{2}{x}^2-2x-2$
23. Find the points of intersection of the following two graphs:
    

    (a)   $y=x^2-6x+8,y=x+2$

    (b)   $y=x^2-x-2,y=x-3$

    Solutions:    (a)  $P_1(1,3),P_2(6,8)$;     (b)  $P(1,-2)$
24. Find the points of intersection of the following two graphs:
    

    (a)   $y=x^2-2x-3,y=-x^2+1$

    (b)   $y=x^2-4x,y=\frac{1}{2}{x}^2-x-\frac{5}{2}$

    Solutions:    (a)  $P_1(-1,0),P_2(2,-3)$;     (b)  $P_1(1,-3),P_2(5,5)$
25. Find all the values of $x$ for which the given function is negative:
    

    (a)   $y=x^2-5x+4$

    (b)   $y=x^2-2x+2$

    Hint:    Draw the graph.
    Solutions:    (a)  $1<x<4$;     (b)  $x$ doesn't exist (this function is always positive)
26. Solve the inequalities:
    

    (a)   $x^2-2x-3⩽0$

    (b)   $x^2-3x+2⩾0$

    Solutions:    (a)  $x\in [-1,3]$;     (b)  $x\in (-\mathrm{\infty },1]\cup [2,\mathrm{\infty })$

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