Solve the quadratic equations:

Exercise 1

4{ x }^{ 2 }-16=0

6{ x }^{ 2 }+3x=0

Exercise 3

4{ x }^{ 2 }+2=0

Exercise 4

{ x }^{ 2 }-2x+1=0

Exercise 5

{ x }^{ 2 }+x+1=0

Exercise 6

{ x }^{ 2 }-4x+4=0

Exercise 7

2x-3=1-2x+{ x }^{ 2 }

Exercise 8

{ x }^{ 2 }+{ (7-x) }^{ 2 }=25

Exercise 9

7{ x }^{ 2 }+21x-28=0

Exercise 10

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

Exercise 11

18=6x+x(x-13)

Exercise 12

6{ x }^{ 2 }-5x+1=0

Exercise 13

{ x }^{ 2 }+{ (x+2) }^{ 2 }=580

Exercise 14

{ x }^{ 2 }-5x-84=0

Exercise 15

4{ x }^{ 2 }-6x+2=0

Exercise 16

{ x }^{ 2 }-\frac { 7 }{ 6 }x+\frac { 1 }{ 3 }=0

 

 

Solution of exercise 1

4{ x }^{ 2 }-16=0

4{ x }^{ 2 }=16

{ x }^{ 2 }=4

{ x }^{ 2 }={ 2 }^{ 2 }

\sqrt { { x }^{ 2 } } =\pm \sqrt { { 2 }^{ 2 } }

x=2 \qquad x=-2

Solution of exercise 2

6{ x }^{ 2 }+3x=0

3x(2x+1)=0

3x=0 \qquad 2x+1=0

x=0 \qquad x=\frac { -1 }{ 2 }

Solution of exercise 3

4{ x }^{ 2 }+2=0

4{ x }^{ 2 }=-2

{ x }^{ 2 }=\frac { -2 }{ 4 }

{ x }^{ 2 }=\frac { -1 }{ 2 }

x=\sqrt { \frac { -1 }{ 2 } } \notin R

 

Solution of exercise 4

{ x }^{ 2 }-2x+1=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-2)\pm \sqrt { { 2 }^{ 2 }-4(1)(1) } }{ 2 }

x=\frac { -(-2)\pm \sqrt { 4-4 } }{ 2 }

x=\frac { -(-2)\pm 0 }{ 2 }

x=\frac { 2 }{ 2 }

x=1

Solution of exercise 5

{ x }^{ 2 }+x+1=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -1\pm \sqrt { { 1 }^{ 2 }-4(1)(1) } }{ 2 }

x=\frac { -1\pm \sqrt { 1-4 } }{ 2a }

x=\frac { -1\pm \sqrt { -3 } }{ 2a }\notin R

Solution of exercise 6

{ x }^{ 2 }-4x+4=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-4)\pm \sqrt { { 4 }^{ 2 }-4(1)(4) } }{ 2(1) }

x=\frac { 4\pm \sqrt { 16-16 } }{ 2 }

x=\frac { 4 }{ 2 }

x=2

Solution of exercise 7

2x-3=1-2x+{ x }^{ 2 }

{ x }^{ 2 }-4x+4=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-4)\pm \sqrt { { -4 }^{ 2 }-4(1)(4) } }{ 2(1) }

x=\frac { 4\pm \sqrt { 16-16 } }{ 2 }

x=\frac { 4\pm 0 }{ 2 }

x=\frac { 4 }{ 2 }

x=2

Solution of exercise 8

{ x }^{ 2 }+{ (7-x) }^{ 2 }=25

{ x }^{ 2 }+49-14x+{ x }^{ 2 }=25

2{ x }^{ 2 }-14x+24=0

{ x }^{ 2 }-7x+12=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { 7\pm \sqrt { { 7 }^{ 2 }-4(1)(12) } }{ 2(1) }

x=\frac { 7\pm \sqrt { 49-48 } }{ 2 }

x=\frac { 7\pm \sqrt { 1 } }{ 2 }

x=\frac { 7\pm 1 }{ 2 }

x=\frac { 7 + 1 }{ 2 } \qquad x=\frac { 7 - 1 }{ 2 }

x=\frac { 8 }{ 2 } \qquad x=\frac { 6 }{ 2 }

x=4 \qquad x=3

Solution of exercise 9

7{ x }^{ 2 }+21x-28=0

{ x }^{ 2 }+3x-4=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -3\pm \sqrt { { 3 }^{ 2 }-4(1)(-4) } }{ 2(1) }

x=\frac { -3\pm \sqrt { 9+16 } }{ 2 }

x=\frac { -3\pm \sqrt { 25 } }{ 2 }

x=\frac { -3\pm 5 }{ 2 }

x=\frac { -3 + 5 }{ 2 } \qquad x=\frac { -3 - 5 }{ 2 }

x=\frac { 2 }{ 2 } \qquad x=\frac { -8 }{ 2 }

x=1 \qquad x=-4

Solution of exercise 10

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

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

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-4)\pm \sqrt { { -4 }^{ 2 }-4(1)(7) } }{ 2 }

x=\frac { 4\pm \sqrt { 16-28 } }{ 2 }

x=\frac { 4\pm \sqrt { -12 } }{ 2 }\notin R

Solution of exercise 11

18=6x+x(x-13)

18=6x+{ x }^{ 2 }-13x

{ x }^{ 2 }-7x-18=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-7)\pm \sqrt { { -7 }^{ 2 }-4(1)(-18) } }{ 2(1) }

x=\frac { 7\pm \sqrt { 49+72 } }{ 2 }

x=\frac { 7\pm \sqrt { 121 } }{ 2 }

x=\frac { 7\pm 11 }{ 2 }

x=\frac { 7 + 11 }{ 2 } \qquad x=\frac { 7 - 11 }{ 2 }

x=\frac { 18 }{ 2 } \qquad x=\frac { -4 }{ 2 }

x=9 \qquad x=-2

 

Solution of exercise 12

6{ x }^{ 2 }-5x+1=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-5)\pm \sqrt { { -5 }^{ 2 }-4(6)(1) } }{ 2(6) }

x=\frac { 5\pm \sqrt { 25-24 } }{ 12 }

x=\frac { 5\pm 1  }{ 12 }

x=\frac { 5 + 1  }{ 12 } \qquad x=\frac { 5 - 1  }{ 12 }

x=\frac { 6  }{ 12 } \qquad x=\frac { 4  }{ 12 }

x=\frac { 1  }{ 2 } \qquad \frac { 1  }{ 3 }

Solution of exercise 13

{ x }^{ 2 }+{ (x+2) }^{ 2 }=580

{ x }^{ 2 }+{ x }^{ 2 }+4x+4=580

2{ x }^{ 2 }+4x-576=0

{ x }^{ 2 }+2x-288=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -2\pm \sqrt { { 2 }^{ 2 }-4(1)(-576)} }{ 2(1) }

x=\frac { -2\pm \sqrt { 4+1152} }{ 2 }

x=\frac { -2\pm \sqrt { 1156} }{ 2 }

x=\frac { -2\pm 34 }{ 2 }

x=\frac { -2 + 34 }{ 2 } \qquad x=\frac { -2 - 34 }{ 2 }

x=\frac { 32 }{ 2 } \qquad x=\frac { -36 }{ 2 }

x= 16 \qquad x=-18

Solution of exercise 14

{ x }^{ 2 }-5x-84=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-5)\pm \sqrt { { -5 }^{ 2 }-4(1)(-84) } }{ 2(1) }

x=\frac { 5\pm \sqrt { 25+336 } }{ 2 }

x=\frac { 5\pm \sqrt { 361 } }{ 2 }

x=\frac { 5\pm 19 }{ 2 }

x=\frac { 5 + 19 }{ 2 } \qquad x=\frac { 5 - 19 }{ 2 }

x=\frac { 24 }{ 2 } \qquad x=\frac { -14 }{ 2 }

x=12 \qquad x=-7

Solution of exercise 15

4{ x }^{ 2 }-6x+2=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-6)\pm \sqrt { { -6 }^{ 2 }-4(4)(2) } }{ 2(4) }

x=\frac { 6\pm \sqrt { 36-32 } }{ 8 }

x=\frac { 6\pm \sqrt { 4 } }{ 8 }

x=\frac { 6\pm 2  }{ 8 }

x=\frac { 6 + 2  }{ 8 } \qquad x=\frac { 6 - 2  }{ 8 }

x=\frac { 8  }{ 8 } \qquad x=\frac { 4  }{ 8 }

x=1 \qquad x=\frac { 1  }{ 2 }

Solution of exercise 16

{ x }^{ 2 }-\frac { 7 }{ 6 }x+\frac { 1 }{ 3 }=0

6{ x }^{ 2 }-7x+2=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-7)\pm \sqrt { { -7 }^{ 2 }-4(6)(2) } }{ 2(6) }

x=\frac { 7\pm \sqrt { 49-48 } }{ 12 }

x=\frac { 7\pm \sqrt { 1 } }{ 12 }

x=\frac { 7\pm 1 }{ 12 }

x=\frac { 7 + 1 }{ 12 } \qquad x=\frac { 7 - 1 }{ 12 }

x=\frac { 8 }{ 12 } \qquad x=\frac { 6 }{ 12 }

x=\frac { 2 }{ 3 } \qquad x=\frac { 1 }{ 2 }

 
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Emma

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