Quadratic function

Quadratic function

Quadratic equations

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)  \(x_1=0,~ x_2=3,~ x_3=-7\)
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)  \(x_1=0,~ x_2=-2,~ x_3=4\)
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)  \(x_1=5,~ x_2=3,~ x_3=-3\)
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)  \(x_1=\frac{2}{3},~ x_2=3\)
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)  \(x_1=4-2\sqrt{2},~ x_2=4+2\sqrt{2}\)
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)  \(x_1=\frac{3-\sqrt{5}}{2},~ x_2=\frac{3+\sqrt{5}}{2}\)
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)  \(\Delta=-39\), no real roots;     (b)  \(\Delta=144\), two real roots;     (c)  \(\Delta=0\), one repeated root
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)  \(\Delta=12,~ x_1=4+\sqrt{3},~ x_2=4-\sqrt{3}\);     (b)  \(\Delta=0,~ x_1=x_2=18\);     (c)  \(\Delta=1,~ x_1=\frac{3}{2},~ x_2=\frac{5}{3}\)
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)  \(m_1=-4,~ m_2=20\)
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)  \(x_1=-\frac{3}{2},~ x_2=-\frac{4}{3}\)
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)  \(x_1=5,\!54,~ x_2=-0,\!541\)
Solve the following equations:

(a)   \(5x-4=\frac{\textstyle 1}{\textstyle x}\)

(b)   \(x-2=\frac{\textstyle 20}{\textstyle x-1}\)

(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)  \(x_1=1,~ x_2=8\)
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)  \(x_1=4,~ x_2=25\)
The product of two positive numbers is 120. The first number is 7 more than the other. Find these two numbers.
Solutions: \(a=15,~ b=8\)
The product of two positive numbers is 403. The first number is 5 more than twice the other number. Find these two numbers.
Solutions: \(a=31,~ b=13\)
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: \(a_1=15,~ b_1=24\); \(a_2=-12,~ b_2=-30\)
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: \(a=60~\mathrm{cm},~ b=35~\mathrm{cm}\)
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 \(51\,400~\mathrm{m}^2\). Determine the area of the first field and the area of the second field.
Solutions: \(a^2=28\,900~\mathrm{m}^2,~ b^2=22\,500~\mathrm{m}^2\)

Graphs of quadratic functions

Draw graphs of the following functions:

(a)   \(f(x)=x^2-4\)

(b)   \(f(x)=x^2-4x\)

(c)   \(f(x)=x^2-4x+3\)
Draw graphs of the following functions:

(a)   \(y=x^2-2x-3\)

(b)   \(y=-x^2+6x-8\)

(c)   \(y=\frac{1}{2}x^2-2x+2\)
Draw graphs of the following functions:

(a)   \(y=2x^2-4x-1\)

(b)   \(y=x^2-4x+5\)

(c)   \(y=2x^2-3x+2\)
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 \(f(x)=ax^2+bx+c\).

(b)   Draw the graph of \(f\).
Solutions:    (a)  \(f(x)=\frac{1}{2}x^2-2x-2\)
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)  \(P(1,-2)\)
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)  \(P_1(1,-3),~ P_2(5,5)\)
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)  \(1\lt x \lt 4\);     (b)  \(x\) doesn't exist (this function is always positive)
Solve the inequalities:

(a)   \(x^2-2x-3\leqslant 0\)

(b)   \(x^2-3x+2\geqslant 0\)
Solutions:    (a)  \(x\in[-1,3]\);     (b)  \(x\in (-\infty,1]\cup[2,\infty)\)

Index