To determine which of the given options are solutions to the equation:
[tex]\[ 2x^2 + 18x = 20 \][/tex]
we need to solve for [tex]\( x \)[/tex] and then check the provided options.
First, we rewrite the equation in standard quadratic form:
[tex]\[ 2x^2 + 18x - 20 = 0 \][/tex]
Next, we solve the quadratic equation. The solutions to the equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 2 \)[/tex], [tex]\( b = 18 \)[/tex], and [tex]\( c = -20 \)[/tex].
Applying these values to the quadratic formula, we get two solutions.
The solutions are:
[tex]\[ x = -10 \][/tex]
[tex]\[ x = 1 \][/tex]
Now, we verify which of the given options are solutions to the equation. The provided options are:
- A. -10
- B. -1
- C. 20
- D. 1
- E. -2
- F. -2
We check each option by substituting it back into the original equation and verifying whether the equation holds.
For A. [tex]\( x = -10 \)[/tex]:
[tex]\[ 2(-10)^2 + 18(-10) = 2(100) - 180 = 200 - 180 = 20 \][/tex]
This satisfies the equation, so [tex]\( x = -10 \)[/tex] is a solution.
For B. [tex]\( x = -1 \)[/tex]:
[tex]\[ 2(-1)^2 + 18(-1) = 2(1) - 18 = 2 - 18 = -16 \][/tex]
This does not satisfy the equation, so [tex]\( x = -1 \)[/tex] is not a solution.
For C. [tex]\( x = 20 \)[/tex]:
[tex]\[ 2(20)^2 + 18(20) = 2(400) + 360 = 800 + 360 = 1160 \][/tex]
This does not satisfy the equation, so [tex]\( x = 20 \)[/tex] is not a solution.
For D. [tex]\( x = 1 \)[/tex]:
[tex]\[ 2(1)^2 + 18(1) = 2(1) + 18 = 2 + 18 = 20 \][/tex]
This satisfies the equation, so [tex]\( x = 1 \)[/tex] is a solution.
For E. [tex]\( x = -2 \)[/tex]:
[tex]\[ 2(-2)^2 + 18(-2) = 2(4) - 36 = 8 - 36 = -28 \][/tex]
This does not satisfy the equation, so [tex]\( x = -2 \)[/tex] is not a solution.
For F. [tex]\( x = -2 \)[/tex] (Again):
This would be the same as option E, so [tex]\( x = -2 \)[/tex] is not a solution.
Therefore, the options that satisfy the equation are:
- A. -10
- D. 1
