Completing the Square, Discriminant & Sum & Product of Roots

Completing the Square, Discriminant and, Sum and Product of Roots

Solving Quadratic Equation by Completing the Square

FA 30.6

Solving Quadratic Equation by Completing the Square

Fill in the Blanks

FA 30.6 #1
Give the number that must be added to complete the square.

 $x^2−6x=8$  

Fill in the Blanks

FA 30.6 #2
Give the number that must be added to complete the square.

 $x^2+10x=3$  

Fill in the Blanks

FA 30.6 #3
Give the number that must be added to complete the square. You may use the "/" sign if your answer is a fraction.

 $x^2+x=5$  

Multiple Choice

FA 30.6 #4
Determine if the given trinomial is a perfect square or not.

 $x^2+6x+36$  

• Perfect Square
  

• Not

Multiple Choice

FA 30.6 #5
Determine if the given trinomial is a perfect square or not.

 $x^2-24x+144$  

• Perfect Square
  

• Not

Multiple Choice

FA 30.6 #6
Solve by completing the square.

 $x^2+8x+6=0$  

•  $x=-2\pm\sqrt{10}$  
  

•  $x=-4\pm\sqrt{10}$  
  

•  $x=3\pm\sqrt{6}$  

•  $x=4\pm\sqrt{3}$  

Multiple Choice

FA 30.6 #7
Solve by completing the square.

 $x^2+14x=32$  

•  $x=2,\ 8$  
  

•  $x=-8,\ 2$  
  

•  $x=2\pm\sqrt{7}$  

•  $x=-16,2$  

Multiple Choice

FA 30.6 #8
Solve by completing the square.

 $2x^2+x-6=0$  

•  $x=-2,\ \frac{3}{2}$  
  

•  $x=-\frac{3}{2},2$  
  

•  $x=-2,\ 3$  

•  $x=2,\ 3$  

end of FA 30.6

Discriminant and Nature of Roots

The roots of the quadratic equation can be rational, irrational, real or imaginary numbers.

Imaginary number  $\left(i\right)$  

The discriminant (or D) =  $b^2-4ac$ .

EXAMPLES

Sum and Product of Roots

Practice Activity

Discriminant and Sum and Product of Roots

Multiple Choice

Practice Activity #1
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal.

 $x^2−x−12=0$  

• D = 0 (real, rational and equal)
  

• D = 49 (real, irrational and unequal)

• D = 49 (real, rational and unequal)
  

• D = -8 (imaginary and unequal)
  

Multiple Choice

Practice Activity #2
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal.

 $5x^2−3x+2=0$  

• D = 0 (real, rational and equal)
  

• D = 31 (real, irrational and unequal)

• D = -31 (imaginary and unequal)
  

• D = 16 (real, rational and unequal
  

Multiple Choice

Practice Activity #3
Find the sum and product of the roots of the given quadratic equation.

 $x^2+2x+1=0$  

• Sum: -2
  
  Product: -1
  

• Sum: 1
  
  Product: 2

• Sum: 2
  
  Product: -1
  

• Sum: -2
  
  Product: 1

Multiple Choice

Practice Activity #4

What is the quadratic equation whose roots are 9 and -4?

•  $x^2-5x-36=0$  
  

•  $x^2-13x-36=0$  
  

•  $x^2+5x-36=0$  
  

•  $x^2-13x+36=0$  
  

Answer Key

Practice Exercises

FA 30.7

Discriminant and, Sum and

Product of Roots

Multiple Choice

FA 30.7 #1
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal
 $6t^2-3=8x$  

• D = 64 ( real, rational and unequal)
  

• D = 0 (real, rational and equal)
  

• D = -48 (imaginary and unequal)
  

• D = 136 (real, irrational and unequal)
  

Multiple Choice

FA 30.7 #2
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal
 $y^2-5y-14=0$  

• D = 81 ( real, rational and unequal)
  

• D = 0 (real, rational and equal)
  

• D = -70 (imaginary and unequal)
  

• D =70 (real, irrational and unequal)
  

Multiple Choice

FA 30.7 #3
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal
 $5x^2-6\left(x-1\right)=0$  

• D = 64 ( real, rational and unequal)
  

• D = 0 (real, rational and equal)
  

• D = -84 (imaginary and unequal)
  

• D =60 (real, irrational and unequal)
  

Multiple Choice

FA 30.7 #4
Use the discriminant and state whether the roots of each equation are real, rational/irrational and equal/unequal, or imaginary and unequal
 $4x^2+20x=-25$  

• D = 100 ( real, rational and unequal)
  

• D = 0 (real, rational and equal)
  

• D = -10 (imaginary and unequal)
  

• D =50 (real, irrational and unequal)
  

Multiple Choice

FA 30.7 #5
Find the sum and product of the roots of the given quadratic equation.
 $2x^2-x-1=0$  

• Sum:  $-\frac{1}{2}$  
  
  Product: - $-\frac{1}{2}$  
  

• Sum:  $\frac{1}{2}$  
  
  Product:  $-\frac{1}{2}$  
  

• Sum:  $-1$  
  
  Product:  $-\frac{1}{2}$  
  

• Sum: 2
  
  Product: 1
  

Multiple Choice

FA 30.7 #6
Find the sum and product of the roots of the given quadratic equation.
 $4y^2+9=0$  

• Sum:  $0$  
  
  Product: - $-\frac{9}{4}$  
  

• Sum:  $\frac{9}{4}$  
  
  Product:  $0$  
  

• Sum:  $-\frac{9}{4}$  
  
  Product:  $0$  
  

• Sum:  $0$  
  
  Product:  $\frac{9}{4}$  
  

Multiple Choice

FA 30.7 #7
Find the quadratic equation with the roots of 5 and -11.

•  $x^2+6x-55=0$  

•  $x^2-6x-55=0$  

•  $x^2+5x-55=0$  

•  $x^2-5x+55=0$  

Multiple Choice

FA 30.7 #8
Find the quadratic equation with the roots of  $\frac{1}{2}$  and 4.

•  $2x^2-9x+4=0$  

•  $2x^2+9x+4=0$  

•  $2x^2-9x+2=0$  

•  $2x^2-9x-2=0$  

end of the test