Determine whether the values −1 and 73 are solutions to the quadratic equation 3x2−4x−4=3 .(1 point) Responses Neither x=−1 nor x=73 is a solution.Neither x equals negative 1 nor x equals Start Fraction 7 over 3 End Fraction is a solution. Only x=−1 is a solution. Only x equals negative 1 is a solution. Only x=73 is a solution. Only x equals Start Fraction 7 over 3 End Fraction is a solution. Both x=−1 and x=73 are solutions.
To determine whether the values −1 and 73 are solutions to the quadratic equation 3x^2−4x−4=3, we substitute each value into the equation and check if it is satisfied.
For x = -1:
3(-1)^2 - 4(-1) - 4 = 3
3(1) + 4 - 4 = 3
3 + 4 - 4 = 3
7 - 4 = 3
3 = 3
Since both sides of the equation are equal when x = -1, it means that x = -1 is a solution.
For x = 73:
3(73)^2 - 4(73) - 4 = 3
3(5329) - 292 - 4 = 3
15987 - 292 - 4 = 3
15691 - 4 = 3
15687 = 3
Since both sides of the equation are not equal when x = 73, it means that x = 73 is not a solution.
Therefore, the correct response is: Neither x equals negative 1 nor x equals Start Fraction 7 over 3 End Fraction is a solution.
Use the table to answer the question. x Substituted Evaluate True Statement? 12 24 36 48 When set off, a certain firework follows the path of the quadratic function h=−2536x2+1623x , where: h= the height of the firework in feet. x= the horizontal distance it travels in feet. To determine how far the firework will travel before reaching the ground, determine which value of x in table is a solution to the equation 0=−2536x2+1623x . (1 point) Responses 12 feet 12 feet 48 feet 48 feet 24 feet 24 feet 36 feet 36 feet
To determine how far the firework will travel before reaching the ground, we need to find the value of x in the given table that is a solution to the equation 0 = −2536x^2 + 1623x.
From the table, we can see that the values of x are 12, 24, 36, and 48.
Let's substitute these values into the equation:
For x = 12:
0 = −2536(12)^2 + 1623(12)
0 = −2536(144) + 1623(12)
0 = −365184 + 19476
0 = −345708
For x = 24:
0 = −2536(24)^2 + 1623(24)
0 = −2536(576) + 1623(24)
0 = −1460736 + 38952
0 = −1421784
For x = 36:
0 = −2536(36)^2 + 1623(36)
0 = −2536(1296) + 1623(36)
0 = −3306816 + 58428
0 = −3248388
For x = 48:
0 = −2536(48)^2 + 1623(48)
0 = −2536(2304) + 1623(48)
0 = −5868544 + 77904
0 = −5790640
From the calculations, we can see that none of the values of x from the table satisfy the equation 0 = −2536x^2 + 1623x.
Therefore, none of the options "12 feet", "24 feet", "36 feet", or "48 feet" are correct.
Ariel is trying to determine if x=−3 is a solution to the quadratic equation −3x2−9x=0. Which explanation demonstrates the correct reasoning?(1 point) Responses Yes, x=−3 is a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? −3 −3(−3)2−9(−3)=0 54=0 TrueYes, x equals negative 3 is a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? negative 3 negative 3 left parenthesis negative 3 right parenthesis squared minus 9 left parenthesis negative 3 right parenthesis equals 0 54 equals 0 True No, x=−3 is not a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? −3 −3(−3)2−9(−3)=0 −54≠0 FalseNo, x equals negative 3 is not a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? negative 3 negative 3 left parenthesis negative 3 right parenthesis squared minus 9 left parenthesis negative 3 right parenthesis equals 0 negative 54 does not equal 0 False No, x=−3 is not a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? −3 −3(−3)2−9(−3)=0 54≠0 FalseNo, x equals negative 3 is not a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? negative 3 negative 3 left parenthesis negative 3 right parenthesis squared minus 9 left parenthesis negative 3 right parenthesis equals 0 54 does not equal 0 False Yes, x=−3 is a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? −3 −3(−3)2−9(−3)=0 0=0 True
To determine if x = -3 is a solution to the quadratic equation -3x^2 - 9x = 0, we substitute it back into the equation and check if it is satisfied.
For x = -3:
-3(-3)^2 - 9(-3) = 0
-3(9) + 27 = 0
-27 + 27 = 0
0 = 0
Since both sides of the equation are equal when x = -3, it means that x = -3 is a solution.
Therefore, the correct explanation is: Yes, x=−3 is a solution because substituting it back into the equation results in the following: x Substituted Evaluate True Statement? −3 −3(−3)2−9(−3)=0 0=0 True.
Show how many solutions are in the solution set for the equation 3x2=21 .(1 point) Responses zero zero infinitely many infinitely many one one two
To find the number of solutions for the equation 3x^2 = 21, we can start by isolating x^2 by dividing both sides of the equation by 3:
x^2 = 21/3
x^2 = 7
Now, to solve for x, we can take the square root of both sides of the equation:
√(x^2) = √(7)
x = ±√(7)
Therefore, the equation 3x^2 = 21 has two solutions: x = √(7) and x = -√(7).
The correct response is: two.
Show how many solutions are in the solution set for 2(x−5)2+2=20.(1 point) Responses two two infinitely many infinitely many zero zero one
To find the solution set for the equation 2(x-5)^2 + 2 = 20, we can start by simplifying the equation:
2(x-5)^2 + 2 = 20
2(x^2 - 10x + 25) + 2 = 20
2x^2 - 20x + 50 + 2 = 20
2x^2 - 20x + 52 = 20
Now, we can rearrange the equation to obtain a quadratic equation in standard form:
2x^2 - 20x + 52 - 20 = 0
2x^2 - 20x + 32 = 0
To find the solutions, we can either factor the quadratic equation or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our equation, a = 2, b = -20, and c = 32. Plugging in the values, we have:
x = (-(-20) ± √((-20)^2 - 4(2)(32)))/(2(2))
x = (20 ± √(400 - 256))/4
x = (20 ± √(144))/4
x = (20 ± 12)/4
This gives us two possible solutions:
x = (20 + 12)/4 = 32/4 = 8
x = (20 - 12)/4 = 8/4 = 2
Therefore, the equation 2(x-5)^2 + 2 = 20 has two solutions: x = 8 and x = 2.
The correct response is: two.
To determine whether the values -1 and 73 are solutions to the quadratic equation 3x^2 - 4x - 4 = 3, we can substitute these values into the equation and check if both sides are equal.
For -1:
Plug -1 into the equation: 3(-1)^2 - 4(-1) - 4 = 3
Simplifying the equation: 3 - (-4) - 4 = 3
Which gives us: 3 + 4 - 4 = 3
Simplifying further: 3 = 3
Since both sides of the equation are equal when we substitute -1, -1 is a solution to the equation.
For 73:
Plug 73 into the equation: 3(73)^2 - 4(73) - 4 = 3
Simplifying the equation: 3 * 5329 - 4 * 73 - 4 = 3
Which gives us: 15987 - 292 - 4 = 3
Simplifying further: 15691 - 4 = 3
Which gives us: 15687 = 3
Since the left side of the equation is not equal to the right side when we substitute 73, 73 is not a solution to the equation.
Therefore, the correct response would be "Only x=-1 is a solution."