Question

task #4: equations and solutions
for each of the equations below, determine whether the given value is a solution or not.

1. ( x + 2 = x^2 + 4 ) at ( x = 2 )

1. ( p + 2 = p^2 - 4 ) at ( p = -2 )

1. ( \frac{a - 5}{a + 5} = 1 ) at ( a = 0 )

1. ( \frac{5 - a}{5 + a} = -1 ) at ( a = 0 )

1. ( 3(x - 8) = 3x - 8 ) at ( x = 0 )

1. which, out of the numbers 0, 1, -1, 2, -2, is/are solution(s) to the equation ( 4x^2 - 4x - 5 = 2x(x + 3) - 1 )?

Explanation:

Step1: Substitute $x=2$ into equation 1

Left side: $2 + 2 = 4$
Right side: $2^2 + 4 = 4 + 4 = 8$

Step2: Check equality for equation 1

$4
eq 8$, so $x=2$ is not a solution.

Step3: Substitute $p=-2$ into equation 2

Left side: $-2 + 2 = 0$
Right side: $(-2)^2 - 4 = 4 - 4 = 0$

Step4: Check equality for equation 2

$0 = 0$, so $p=-2$ is a solution.

Step5: Substitute $a=0$ into equation 3

Left side: $\frac{0 - 5}{0 + 5} = \frac{-5}{5} = -1$
Right side: $1$

Step6: Check equality for equation 3

$-1
eq 1$, so $a=0$ is not a solution.

Step7: Substitute $a=0$ into equation 4

Left side: $\frac{5 - 0}{5 + 0} = \frac{5}{5} = 1$
Right side: $-1$

Step8: Check equality for equation 4

$1
eq -1$, so $a=0$ is not a solution.

Step9: Substitute $x=0$ into equation 5

Left side: $3(0 - 8) = 3(-8) = -24$
Right side: $3(0) - 8 = 0 - 8 = -8$

Step10: Check equality for equation 5

$-24
eq -8$, so $x=0$ is not a solution.

Step11: Simplify equation 6 first

$$\begin{align*} 4x^2 - 4x - 5 &= 2x(x+3) - 1\\ 4x^2 - 4x - 5 &= 2x^2 + 6x - 1\\ 4x^2 - 2x^2 - 4x - 6x - 5 + 1 &= 0\\ 2x^2 - 10x - 4 &= 0\\ x^2 - 5x - 2 &= 0 \end{align*}$$

Step12: Test each value in simplified equation 6

• For $x=0$: $0^2 - 5(0) - 2 = -2

eq 0$

• For $x=1$: $1^2 - 5(1) - 2 = 1 - 5 - 2 = -6

eq 0$

• For $x=-1$: $(-1)^2 - 5(-1) - 2 = 1 + 5 - 2 = 4

eq 0$

• For $x=2$: $2^2 - 5(2) - 2 = 4 - 10 - 2 = -8

eq 0$

• For $x=-2$: $(-2)^2 - 5(-2) - 2 = 4 + 10 - 2 = 12

eq 0$
None of the values satisfy the equation.

Answer:

1. $x=2$ is not a solution.
2. $p=-2$ is a solution.
3. $a=0$ is not a solution.
4. $a=0$ is not a solution.
5. $x=0$ is not a solution.
6. None of the numbers 0, 1, -1, 2, -2 are solutions.