Solve the following equation
(x−5)(x+3)=−7

. 3

Step-by-step answer

09.04.2023, solved by verified expert
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Answer:

x=4, -2

Step-by-step explanation:

Given equation 

(x-5) (x+3) =-7

On solving 

(x^2 +3x -5x -15) = -7

Combining the similar terms 

x^2 -2x -8 = 0

Using middle term splitting method

x^2 -4x + 2x -8 = 0

x(x-4) + 2(x-4) = 0

(x-4) (x+2) = 0

Therefore x= 4 or x= -2 

Solution is 

x=4 , -2

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Mathematics
Step-by-step answer
P Answered by PhD

Following are the calculation to the given points:

For point 1:

\bold{8x^2+26x+15}\\\\\bold{8x^2+(20+6)x+15}\\\\\bold{8x^2+20x+6x+15}\\\\\bold{4x(2x+5)+3(2x+5)}\\\\\bold{(2x+5)(4x+3)}\\\\

Therefore, the answer is "Option D".

For point 2:

\bold{x^2 + 5x - 24}\\\\\bold{x^2 + (8-3)x - 24}\\\\\bold{x^2 + 8x-3x - 24}\\\\\bold{x(x + 8)-3(x +8)}\\\\\bold{(x + 8) (x-3)}\\\\

Therefore, the answer is "Option A".

For point 3:

\bold{8x2 - 50}\\\\\bold{2(4x2 - 25)}\\\\\bold{2((2x)^2 - 5^2)}\\\\\therefore \ \ x^2-y^2=(x+y) (x-y)\\\\\bold{2((2x - 5)(2x+5)}\\\\

Therefore, the answer is "Option D".

For point 4:

\bold{3x^2 + 7x + 2}\\\\\bold{3x^2 +(6+1)x + 2}\\\\\bold{3x^2 +6x+1x + 2}\\\\\bold{3x(x +2)+1(x + 2)}\\\\\bold{(x +2) (3x+1)}\\\\

Therefore, the answer is "Option D".

For point 5:

\bold{x^2 - 5x = 14}\\\\\bold{x^2 - 5x -14= 0}\\\\\bold{x^2 - (7-2)x -14= 0}\\\\\bold{x^2 -7x+2x -14= 0}\\\\\bold{x(x -7)+2(x -7)= 0}\\\\\bold{(x -7) (x+2)= 0}\\\\\bold{x -7=0\ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ x+2= 0}\\\\\bold{x =7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x= -2}\\\\

Therefore, the answer is "Option D".

For point 6:

\bold{x^2 + 5x - 50}\\\\\bold{x^2 + (10-5)x - 50}\\\\\bold{x^2 + 10x-5x - 50}\\\\\bold{x(x+ 10)-5(x +10)}\\\\\bold{(x+ 10)(x -5)}\\\\

Therefore, the answer is "Option B".

For point 7:

\bold{16x^2 + 24x + 9}\\\\\bold{16x^2 + (12+12)x + 9}\\\\\bold{16x^2 + 12x+12x + 9}\\\\\bold{4x(4x +3)+3(4x + 3)}\\\\\bold{(4x +3)(4x + 3)}\\\\\bold{(4x +3)^2}\\\\

Therefore, the answer is "Option A".

For point 8:

\bold{32x^2 - 50}\\\\\bold{2(16x^2 - 25)}\\\\\bold{2((4x)^2 - (5)^2)}\\\\\bold{2((4x-5)(4x+5))}\\\\

Therefore, the answer is "Option B".

For point 9:

\bold{x^2 + 7x - 18}\\\\\bold{x^2 + (9-2)x - 18}\\\\\bold{x^2 + 9x-2x - 18}\\\\\bold{x(x+ 9)-2(x +9)}\\\\\bold{(x+ 9)(x-2)}\\\\

Therefore, the answer is "Option C".

For point 10:

\bold{x^2 + 5x - 14}\\\\\bold{x^2 + (7-2)x - 14}\\\\\bold{x^2 + 7x-2x - 14}\\\\\bold{x(x+7)-2(x +7)}\\\\\bold{(x+7)(x -2)}\\\\

Therefore, the answer is "Option C".

For point 11:

\bold{25x^2 - 64}\\\\\bold{(5x)^2 - (8)^2}\\\\\bold{(5x-8)(5x+8)}\\\\

Therefore, the answer is "Option C".

For point 12:

\bold{x^2 - 81}\\\\\bold{x^2 - 9^2}\\\\\bold{(x - 9)(x+9)}\\\\

Therefore, the answer is "Option A".

For point 13:

\bold{8x^2 + 16x + 8 = 0}\\\\\bold{8(x^2 + 2x + 1) = 0}\\\\\bold{(x^2 + 2x + 1) = 0}\\\\\bold{(x^2 + x+x + 1) = 0}\\\\\bold{(x(x +1)1(x + 1)) = 0}\\\\\bold{(x +1)(x + 1) = 0}\\\\\bold{(x +1)^2 = 0}\\\\\bold{x +1 = 0}\\\\\bold{x=-1}\\\\

Therefore, the answer is "Option B".

For point 14:

\bold{x^2 -x -12 = 0}\\\\\bold{x^2 -(4-3)x -12 = 0}\\\\\bold{x^2 -4x+3x -12 = 0}\\\\\bold{x(x -4)+3(x -4) = 0}\\\\\bold{(x -4) (x+3) = 0}\\\\\bold{(x -4)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x+3) = 0}\\\\\bold{x =4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = -3}\\\\

Therefore, the answer is "Option C".

For point 15:

\bold{6x^2 + 8x - 28 = 2x^2 + 4}\\\\\bold{6x^2 + 8x - 28 - 2x^2 - 4=0}\\\\\bold{4x^2 + 8x - 32=0}\\\\\bold{4(x^2 + 2x - 8)=0}\\\\\bold{(x^2 + 2x - 8)=0}\\\\\bold{(x^2 + (4-2)x - 8)=0}\\\\\bold{(x^2 + 4x-2x - 8)=0}\\\\\bold{(x(x + 4)-2(x +4))=0}\\\\\bold{(x + 4)(x -2)=0}\\\\\bold{(x + 4)=0\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x -2)=0}\\\\\bold{x =-4\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x =2}\\\\

Therefore, the answer is "Option B".

For point 14:

\bold{6z^2 + 18z}\\\\\bold{6z(z + 3)}\\\\

Therefore, the answer is "Option B".

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1) Which is a binomial factor of 8x2 + 26x + 15? A) x − 5  B) 2x − 5  C) 2x + 3  D) 4x + 3  2) Facto
1) Which is a binomial factor of 8x2 + 26x + 15? A) x − 5  B) 2x − 5  C) 2x + 3  D) 4x + 3  2) Facto
Mathematics
Step-by-step answer
P Answered by PhD

Following are the calculation to the given points:

For point 1:

\bold{8x^2+26x+15}\\\\\bold{8x^2+(20+6)x+15}\\\\\bold{8x^2+20x+6x+15}\\\\\bold{4x(2x+5)+3(2x+5)}\\\\\bold{(2x+5)(4x+3)}\\\\

Therefore, the answer is "Option D".

For point 2:

\bold{x^2 + 5x - 24}\\\\\bold{x^2 + (8-3)x - 24}\\\\\bold{x^2 + 8x-3x - 24}\\\\\bold{x(x + 8)-3(x +8)}\\\\\bold{(x + 8) (x-3)}\\\\

Therefore, the answer is "Option A".

For point 3:

\bold{8x2 - 50}\\\\\bold{2(4x2 - 25)}\\\\\bold{2((2x)^2 - 5^2)}\\\\\therefore \ \ x^2-y^2=(x+y) (x-y)\\\\\bold{2((2x - 5)(2x+5)}\\\\

Therefore, the answer is "Option D".

For point 4:

\bold{3x^2 + 7x + 2}\\\\\bold{3x^2 +(6+1)x + 2}\\\\\bold{3x^2 +6x+1x + 2}\\\\\bold{3x(x +2)+1(x + 2)}\\\\\bold{(x +2) (3x+1)}\\\\

Therefore, the answer is "Option D".

For point 5:

\bold{x^2 - 5x = 14}\\\\\bold{x^2 - 5x -14= 0}\\\\\bold{x^2 - (7-2)x -14= 0}\\\\\bold{x^2 -7x+2x -14= 0}\\\\\bold{x(x -7)+2(x -7)= 0}\\\\\bold{(x -7) (x+2)= 0}\\\\\bold{x -7=0\ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ x+2= 0}\\\\\bold{x =7\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x= -2}\\\\

Therefore, the answer is "Option D".

For point 6:

\bold{x^2 + 5x - 50}\\\\\bold{x^2 + (10-5)x - 50}\\\\\bold{x^2 + 10x-5x - 50}\\\\\bold{x(x+ 10)-5(x +10)}\\\\\bold{(x+ 10)(x -5)}\\\\

Therefore, the answer is "Option B".

For point 7:

\bold{16x^2 + 24x + 9}\\\\\bold{16x^2 + (12+12)x + 9}\\\\\bold{16x^2 + 12x+12x + 9}\\\\\bold{4x(4x +3)+3(4x + 3)}\\\\\bold{(4x +3)(4x + 3)}\\\\\bold{(4x +3)^2}\\\\

Therefore, the answer is "Option A".

For point 8:

\bold{32x^2 - 50}\\\\\bold{2(16x^2 - 25)}\\\\\bold{2((4x)^2 - (5)^2)}\\\\\bold{2((4x-5)(4x+5))}\\\\

Therefore, the answer is "Option B".

For point 9:

\bold{x^2 + 7x - 18}\\\\\bold{x^2 + (9-2)x - 18}\\\\\bold{x^2 + 9x-2x - 18}\\\\\bold{x(x+ 9)-2(x +9)}\\\\\bold{(x+ 9)(x-2)}\\\\

Therefore, the answer is "Option C".

For point 10:

\bold{x^2 + 5x - 14}\\\\\bold{x^2 + (7-2)x - 14}\\\\\bold{x^2 + 7x-2x - 14}\\\\\bold{x(x+7)-2(x +7)}\\\\\bold{(x+7)(x -2)}\\\\

Therefore, the answer is "Option C".

For point 11:

\bold{25x^2 - 64}\\\\\bold{(5x)^2 - (8)^2}\\\\\bold{(5x-8)(5x+8)}\\\\

Therefore, the answer is "Option C".

For point 12:

\bold{x^2 - 81}\\\\\bold{x^2 - 9^2}\\\\\bold{(x - 9)(x+9)}\\\\

Therefore, the answer is "Option A".

For point 13:

\bold{8x^2 + 16x + 8 = 0}\\\\\bold{8(x^2 + 2x + 1) = 0}\\\\\bold{(x^2 + 2x + 1) = 0}\\\\\bold{(x^2 + x+x + 1) = 0}\\\\\bold{(x(x +1)1(x + 1)) = 0}\\\\\bold{(x +1)(x + 1) = 0}\\\\\bold{(x +1)^2 = 0}\\\\\bold{x +1 = 0}\\\\\bold{x=-1}\\\\

Therefore, the answer is "Option B".

For point 14:

\bold{x^2 -x -12 = 0}\\\\\bold{x^2 -(4-3)x -12 = 0}\\\\\bold{x^2 -4x+3x -12 = 0}\\\\\bold{x(x -4)+3(x -4) = 0}\\\\\bold{(x -4) (x+3) = 0}\\\\\bold{(x -4)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x+3) = 0}\\\\\bold{x =4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = -3}\\\\

Therefore, the answer is "Option C".

For point 15:

\bold{6x^2 + 8x - 28 = 2x^2 + 4}\\\\\bold{6x^2 + 8x - 28 - 2x^2 - 4=0}\\\\\bold{4x^2 + 8x - 32=0}\\\\\bold{4(x^2 + 2x - 8)=0}\\\\\bold{(x^2 + 2x - 8)=0}\\\\\bold{(x^2 + (4-2)x - 8)=0}\\\\\bold{(x^2 + 4x-2x - 8)=0}\\\\\bold{(x(x + 4)-2(x +4))=0}\\\\\bold{(x + 4)(x -2)=0}\\\\\bold{(x + 4)=0\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (x -2)=0}\\\\\bold{x =-4\ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x =2}\\\\

Therefore, the answer is "Option B".

For point 14:

\bold{6z^2 + 18z}\\\\\bold{6z(z + 3)}\\\\

Therefore, the answer is "Option B".

Learn more:

link


1) Which is a binomial factor of 8x2 + 26x + 15? A) x − 5  B) 2x − 5  C) 2x + 3  D) 4x + 3  2) Facto
1) Which is a binomial factor of 8x2 + 26x + 15? A) x − 5  B) 2x − 5  C) 2x + 3  D) 4x + 3  2) Facto
Mathematics
Step-by-step answer
P Answered by PhD

1. 3x^2-4x=x(3x-4) - A.

2. x^2 + 21x +20=(x-x_1)(x-x_2)

Find the roots:

D=21^2-4\cdot 20=441-80=361, \ \sqrt{D}=19,\\ \\x_1=\dfrac{-21-19}{2}=-20, \ x_2=\dfrac{-21+19}{2}=-1,

then

x^2 + 21x +20=(x+20)(x+1) - C.

3. 4x^2 -9=(2x)^2-3^2=(2x-3)(2x+3) - D.

4. 12x^3-36x^2=12x^2(x-3) - C.

5. x^2 -5x -6=(x-x_1)(x-x_2)

Find the roots:

D=(-5)^2-4\cdot (-6)=25+24=49, \ \sqrt{D}=7,\\ \\x_1=\dfrac{5-7}{2}=-1, \ x_2=\dfrac{5+7}{2}=6,

then

x^2 -5x -6=(x-6)(x+1) and the width of the lawn is x-6 - A.

6. Since x^2 + 5x -24=(x+8)(x-3) the length and width are x+8 and x-3 - A.

7. 32 -8z^2=8(4-z^2)=8(2^2-z^2)=8(2-z)(2+z) - B.

8. 12x^2=2\cdot 2\cdot 3\cdot x\cdot x, \\24x^2y^2=2\cdot 2\cdot 2\cdot3\cdot x\cdot x\cdot y\cdot y , \\ 46xy=2\cdot 23\cdot x\cdot y.

So the greatest common divisor is 2\cdot x=2x - D.

9. 2x^2 -3x -35=2(x-x_1)(x-x_2)

Find the roots:

D=(-3)^2-4\cdot (-35)\cdot 2=9+280=289, \ \sqrt{D}=17,\\ \\x_1=\dfrac{3-17}{2\cdot 2}=-\dfrac{7}{2}, \ x_2=\dfrac{3+17}{2\cdot 2}=5,

then

2x^2 -3x -35=2(x+\dfrac{7}{2})(x-5)=(2x+7)(x-5) - A.

10. 81x^2 + 36x + 4=(9x)^2+2\cdot 9x\cdot 2+2^2=(9x+2)^2 - B.

11. 18x^2+ 69x +60=3(6x^2+23x+20)=3\cdot 6(x+\dfrac{5}{2})(x+\dfrac{4}{3})=(6x+15)(3x+4), the length is 6x+15.

12. (x^2 +2x)(5x -3) =x^2\cdot 5x-x^2\cdot 3+2x\cdot 5x-2x\cdot 3=5x^3-3x^2+10x^2-6x=5x^3+7x^2-6x - C.

13. 30g^5 +24g^3h- 35g^2h^2 - 28h^3=(30g^5 +24g^3h)-(35g^2h^2+ 28h^3)=6g^3(5g^2+4h)-7h^2(5g^2+4h)=(5g^2+4h)(6g^3-7h^2).

14. x^2 -16=(x-4)(x+4) - B.

15. 81p^2 + 90p +25=(9p)^2+2\cdot 9p\cdot 5+5^2=(9p+5)^2, the length of one side is 9p+5.

Mathematics
Step-by-step answer
P Answered by PhD

1. 3x^2-4x=x(3x-4) - A.

2. x^2 + 21x +20=(x-x_1)(x-x_2)

Find the roots:

D=21^2-4\cdot 20=441-80=361, \ \sqrt{D}=19,\\ \\x_1=\dfrac{-21-19}{2}=-20, \ x_2=\dfrac{-21+19}{2}=-1,

then

x^2 + 21x +20=(x+20)(x+1) - C.

3. 4x^2 -9=(2x)^2-3^2=(2x-3)(2x+3) - D.

4. 12x^3-36x^2=12x^2(x-3) - C.

5. x^2 -5x -6=(x-x_1)(x-x_2)

Find the roots:

D=(-5)^2-4\cdot (-6)=25+24=49, \ \sqrt{D}=7,\\ \\x_1=\dfrac{5-7}{2}=-1, \ x_2=\dfrac{5+7}{2}=6,

then

x^2 -5x -6=(x-6)(x+1) and the width of the lawn is x-6 - A.

6. Since x^2 + 5x -24=(x+8)(x-3) the length and width are x+8 and x-3 - A.

7. 32 -8z^2=8(4-z^2)=8(2^2-z^2)=8(2-z)(2+z) - B.

8. 12x^2=2\cdot 2\cdot 3\cdot x\cdot x, \\24x^2y^2=2\cdot 2\cdot 2\cdot3\cdot x\cdot x\cdot y\cdot y , \\ 46xy=2\cdot 23\cdot x\cdot y.

So the greatest common divisor is 2\cdot x=2x - D.

9. 2x^2 -3x -35=2(x-x_1)(x-x_2)

Find the roots:

D=(-3)^2-4\cdot (-35)\cdot 2=9+280=289, \ \sqrt{D}=17,\\ \\x_1=\dfrac{3-17}{2\cdot 2}=-\dfrac{7}{2}, \ x_2=\dfrac{3+17}{2\cdot 2}=5,

then

2x^2 -3x -35=2(x+\dfrac{7}{2})(x-5)=(2x+7)(x-5) - A.

10. 81x^2 + 36x + 4=(9x)^2+2\cdot 9x\cdot 2+2^2=(9x+2)^2 - B.

11. 18x^2+ 69x +60=3(6x^2+23x+20)=3\cdot 6(x+\dfrac{5}{2})(x+\dfrac{4}{3})=(6x+15)(3x+4), the length is 6x+15.

12. (x^2 +2x)(5x -3) =x^2\cdot 5x-x^2\cdot 3+2x\cdot 5x-2x\cdot 3=5x^3-3x^2+10x^2-6x=5x^3+7x^2-6x - C.

13. 30g^5 +24g^3h- 35g^2h^2 - 28h^3=(30g^5 +24g^3h)-(35g^2h^2+ 28h^3)=6g^3(5g^2+4h)-7h^2(5g^2+4h)=(5g^2+4h)(6g^3-7h^2).

14. x^2 -16=(x-4)(x+4) - B.

15. 81p^2 + 90p +25=(9p)^2+2\cdot 9p\cdot 5+5^2=(9p+5)^2, the length of one side is 9p+5.

Mathematics
Step-by-step answer
P Answered by Master

Part A

1. Third degree polynomial:  f(x) = ax^3 + bx^2 + cx + d

The function can have as many as 3 zeroes only, but it can have 4 intercepts if you include the y-intercept.

2.  g(x) = (x + 2)(x - 1)(x - 2) =x^3-x^2-4x+4

Zeroes at: x = -2, x = 1 and x = 2

y-intercept at (0, 4)

End behavior:

x \rightarrow-\infty, f(x) \rightarrow-\infty\\\\x \rightarrow\infty, f(x) \rightarrow\infty

3. see attached

Part B

4. f(x)=(x-2)(x-5)=x^2-7x+10

Direction: opens upwards

y-intercept: (0, 10)

zeros: x = 2, x = 5

5. see attached

**unfortunately I am unable to answer any additional questions as has a 5-question rule**


Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get
Ray and Kelsey have summer internships at an engineering firm. As part of their internship, they get
Mathematics
Step-by-step answer
P Answered by PhD

Step-by-step explanation:

1. x^2 -2x + 8x -16 = x^2 +6x -16

2. x^2 -12x +6x - 72 =  x^2 -6x - 72

here's one more for you

7. x^2 -3x +3x - 9 = x^2 -9

9. x^2 +12x-3x-36 = x^2 +9x -36

multiply the terms and then combine like terms.

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