Quadratics

Quadratics 



1.1 Solving quadratic equation

A) Find the zeros of the following

1. $(x-3)(x+2)=0$

2. $(2x+5)(2x+5)=0$

3. $x(x+2)=0$

4. $12(x+3)(2x+1)=0$

B) Find the roots of the following equations:

1. $x^{2}+5x+6=0$

2. $x^{2}+x=6$

3. $x^{2}-x-6=0$

4. $4x^{2}-9x+2=0$

5. $x^{2}+4x=5$

6. $x^{2}+x-72=0$

7. $x^{2}-2x-3=0$

8. $x^{2}+5x+4=0$

9. $x^{2}-6x+5=0$

10. $x^{2}+3x-10=0$

11. $x^{2}-5x=14$

12. $x^{2}+14=9x$

1.2 The process of completing the square

A) Write the following expression in the form $a(x+p)^{2} + q$

1. $x^{2}-2x+3$

2. $x^{2}+4x+1$

3. $–x^{2}+2x+2$

4. $-x^{2}+8x-19$

5. $-2x^{2}+5x-3$

6. $2x^{2}-3x-2$

B) Solve the following equations, by completing the square, give your answer in surd form

1. $x^{2}+8x=1$

2. $x^{2}-2x-2=0$

3. $x^{2}+x-1=0$

4. $x^{2}-x=3$

5. $4x^{2}+x-1=0$

6. $2x^{2}-3x-4=0$

7. $3x^{2}+x-1=0$

8. $2x^{2}+4x=7$

1.3 Solving quadratic equations by using the formula

A) Solve the equations by using the formula, give your answers in surd form

1. $x^{2}+4x+2=0$

2. $2x^{2}-x-2=0$

3. $1+x-3x^{2}=0$

4. $2x^{2}-x=5$

B) Find, correct to 3 decimal places, the roots of equations

1. $5x^{2}+9x+2=0$

2. $2x^{2}-7x+4=0$

3. $8x-x^{2}=1$

4. $x^{2}-3x=1$

1.4 Locate the vertex of the functions

A) For each following, use the method of completing the square find the maximum or minimum value of $y$ and the value of $x$ at which it occurs

1. $y=x^{2}+2x-5$

2. $y=x^{2}+3x+8$

3. $y=x^{2}-7x+15$

4. $y=2x^{2}+10x-5$

5. $y=3x^{2}+6x+14$

6. $y=4x^{2}+x-7$

7. $y=20+6x-x^{2}$

8. $y=4-3x-x^{2}$

9. $y=-2+4x-x^{2}$

10. $y=6-x-x^{2}$

11. $y=3+2x-2x^{2}$

12. $y=7-3x-4x^{2}$

B) By completing the square, sketch the graphs of these quadratics

1. $y=x^{2}-6x+8$

2. $y=2x^{2}+2x-15$

3. $y=3-4x-4x^{2}$

4. $y=2x^{2}-4x+5$

5. $y=8+2x-x^{2}$

6. $y=4x^{2}-20x+21$

1.5 The properties of the roots of quadratic equations

A) Without solving the equation, determine the nature of the roots of each equation

1. $x^{2}-6x+4$

2. $3x^{2}+4x+2$

3. $2x^{2}-5x+3$

4. $x^{2}-6x+9$

5. $4x^{2}+12x+9$

6. $x^{2}+4x-8$

7. $x^{2}-ax+a^{2}$

8. $x^{2}+2ax+a^{2}$

B)

1. Show that $x^{2}+3x+5>0$ for all values of  $x$.

2. Show that $3-2x-x^{2}<0$ for $x>1$.

3. If the roots of $3x^{2}+kx+12=0$ are equal, find $k$.

4. The roots of $x^{2}+px+(p+1)=0$ are equal, find $p$.

5. Find the possible values of the constant $p$ given that the equation $px^{2}+(8-p)x+1=0$ has the repeated root.

6. Given that the equation $x^{2}+3bx+(4b+1)=0$ has a repeated root, find the possible values of the constant $b$.

7. Prove that the roots of the equation $kx^{2}+(2k+4)x+8=0$ are real for all values of $k$.

8. Find the relationship between $p$ and $q$ if the roots of $px^{2}+qx+1=0$ are equal.

C)

1. Find the range of values of $a$ for which $(2-3a)x^{2}+(4-a)x+2=0$ has no real roots.

2. Find the values of $k$ for which the line $y=x+k$ is a tangent to the curve $x^{2}+xy+2=0$.

3. Find the real values of $k$ for which the equation $x^{2}+(k+1)x+k^{2}=0$ has real roots.

4. The equation $x^{2}+x(2x+p)+3=0$ has equal roots, find the possible values of $p$.

5. If the equation $2kx^{2}+4x-3=0$ has equal roots, find the value of $k$.

6. If $y=kx+x+19$ is the tangent to the curve $y=x^{2}+5k$. Find the possible values of $k$.

7. Find the range of values of $m$ for which $mx^{2}+3x+5=0$ can be solved.

1.6 Solve linear and quadratics inequalities

A) Solve the following inequalities

1. $3x+2>x+8$

2. $3x-2<4x-7$

3. $1<2x-3<9$

4. $(x-3)(x+1)<0$

5. $x^{2}-3>2x$

6. $-x^{2}+4x-3<0$

7. $x-4<3-x$

8. $x+3<3x-5$

9. $2(3x-5)>6$

10. $3(3-2x)<2(3+x)$

Find the ranges of values of $x$ that satisfy the following inequalities

1. $(x-2)(x-1)>0$

2. $(x+3)(x-5) \ge 0$

3. $x^{2}-4x>3$

4. $4x^{2}<1$

5. $5x^{2}>3x+2$

6. $(x-1)^{2}>9$

7. $(3-2x)(x+5) \le 0$

8. $(1-x)(4-x)>x+11$

1.7 Solving simultaneous equations

Solve the following sets of equations

1. $5x+2y=7$,$2x+y=2$

2. $x-2y=5$,$3x+y=8$

3. $2x+3y=1$,$3x+y=5$

4. $3x-2y=5$,$2x+y=8$

5. $3x+5y=1$,$2x+3y=0$

6. $5x+4y=1$,$7x+5y=2$

7. $x^{2}+y^{2}=5$,$y-x=1$

8. $3x^{2}-y^{2}=3$,$2x-y=1$

9. $y^{2}-xy=14$,$y=3-x$

10. $xy=2$,$x+y-3=0$

11. $xy+y^{2}=2$,$2x+y=3$

12. $xy+x=-3$,$2x+5y=8$

1.8 Equation reducible to quadratic equation

A) Solve each equations for $x$.

1. $x^{4}-13x^{2}+36=0$

2. $x^{4}-2x^{2}-3=0$

3. $x^{6}-28x^{3}+27=0$

4. $x^{6}+5x^{3}-24=0$

5. $x-5\sqrt{x}+6=0$

6. $x-6\sqrt{x}+5=0$

7. $x^{4}+x^{2}=12$

8. $x=4\sqrt{x}-3$

9. $x^{8}+16=17x^{4}$

10. $x^{6}=8+2x^{3}$

11. $8\sqrt{x}=15+x$

12. $65x^{4}=16+4x^{8}$

13. $x^{4}-26x^{2}+25=0$

14. $x^{6}-9x^{3}+8=0$

15. $x^{6}+7x^{3}-8=0$

16. $e^{2x}+25=26e^{2x}$

B) Solve each of these equations for $x$

1. $(x+3)^{2}-5(x+3)+4=0$

2. $(3x-1)^{2}+6(3x-1)=7$

3. Solve $y^{2}-7y+10=0$. Hence, find the solution to $(x^{2}+1)^{2}-7(x^{2}+1)+10=0$

4. Solve $y^{2}-5y-14=0$. Hence, find the solutions to $(x^{3}-1)^{2}-5(x^{3}-1)-14=0$

5. Solve the equation $x^{4}-4x^{2}+3=0$

6. Solve the equation $3y^{4}-2y^{2}-7=0$

7. Find where the line $y=x+3$ crosses the circle $x^{2}+y^{2}=20$

8. A recrangle tile has length $x$cm and breath $(6-x)$cm. Given that the area of the tile must be at least $5$cm$^{2}$, form a quadratic inequalities in $x$ and hence find the set of possible values of $x$.

1.9 Mixed Exercise

1. Find the range of values of $k$ for which the quadratic equation $x^{2}+(k-4)x+(k-1)=0$ has real distinct roots.

2. The quadratic equation $x^{2}+6x+1=k(x^{2}+1)$ has equal roots. Find the possible values of the constant $k$.

3. Solve the simultaneous equation $y=x-2$, $y^{2}=x$

4. Show that the elimination of $x$ from the simultaneous equations $x-2y=1$,$3xy-y^{2}=8$ produces the equation $5y^{2}+3y-8=0$. Solve this equation and hence find the pairs $(x,y)$ for which the simultaneous equations are satisfied.

5. Solve the simultaneous equations $x+y=2$,$x^{2}+2y^{2}=11$

6. Find the set of values of $x$ for which $2x(x-3)>(x+2)(x-3)$

7. Find the set of values of $x$ for which $2(x^{2}-5)<x^{2}+6$

8. Find the set of values of $x$ for which $x^{2}-x-12>0$

9. Find the values of $k$ for which the line $x+3y=k$ and the curve $y^{2}=2x+3$ do not intersect

10. Find the values of $m$ for which the line $y=mx-9$ is a tangent to the curve $x^{2}=4y$

11. Find the values of $k$ for which the line $y=kx-2$ meets the curve $y^{2}=4x-x^{2}$

12. Find the value of $m$ for which the line $y=mx-3$ is a tangent to the curve $y=x+\frac{1}{x}$ and find the $x$-coordinate of the point at which this tangent touches the curve.

13. Find the value of $c$ and of $d$ for which {$x:-5<x<3$} is the solution set of $x^{2}+cx<d$

14. Find the values of the constant $c$ for which the line $2y=x+c$ is tangent to the curve $y=2x+\frac{6}{x}$