Determine whether the values −1

Question

and 73
are solutions to the quadratic equation 3x2−4x−4=3
.(1 point)
Responses

Only x=73
is a solution.
Only 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.

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.

Both x=−1
and x=73
are solutions.

Answer 1

To determine if the values $x=-1$ and $x=73$ are solutions to the quadratic equation $3x^2-4x-4=3$, we first rewrite the equation in standard form.

Starting with the original equation: $$3x^2-4x-4=3$$

Subtracting 3 from both sides gives: $$3x^2-4x-7=0$$

Now we will test each value:

Testing $x=-1$:

Substituting $-1$ into the equation: $$3(-1{)}^2-4(-1)-7=0$$ Calculating each term: $$3(1)+4-7=3+4-7=0$$ So, $x=-1$ is indeed a solution.

Testing $x=73$:

Substituting $73$ into the equation: $$3(73{)}^2-4(73)-7=0$$ Calculating ${73}^2$: $$3(5329)-292-7=0$$ Calculating: $$15987-292-7=15987-299=15688\ne 0$$ So, $x=73$ is not a solution.

Conclusion

The only solution is $x=-1$.

Thus, the response is: Only $x=-1$ is a solution.

Answer 2

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

48 feet
48 feet

24 feet
24 feet

36 feet
36 feet

12 feet

Answer 3

To determine how far the firework will travel before reaching the ground, we need to find the value of $x$ that makes the height $h$ equal to zero in the quadratic equation:

$$0=-2536x^2+1623x$$

This can be factored out as follows:

$$0=x(-2536x+1623)$$

Setting each factor to zero gives us two solutions:

1. $x=0$
2. $-2536x+1623=0$

Solving the second equation for $x$: $$-2536x+1623=0⟹2536x=1623⟹x=\frac{1623}{2536}$$

Now we can calculate $\frac{1623}{2536}$: $$x\approx 0.639$$

Since this value of $x$ is much smaller than the values given in the table, we need to evaluate the function $-2536x^2+1623x$ at $x=12,24,36,$ and $48$ to see when $h=0$.

Evaluating the Function at the Given Values:

1. For $x=12$: $$h=-2536({12}^2)+1623(12)=-2536(144)+1623(12)$$ $$h=-364224+19476=-344748\text{ (not a solution)}$$

2. For $x=24$: $$h=-2536({24}^2)+1623(24)=-2536(576)+1623(24)$$ $$h=-1458304+38952=-1419352\text{ (not a solution)}$$

3. For $x=36$: $$h=-2536({36}^2)+1623(36)=-2536(1296)+1623(36)$$ $$h=-3282176+58428=-3223748\text{ (not a solution)}$$

4. For $x=48$: $$h=-2536({48}^2)+1623(48)=-2536(2304)+1623(48)$$ $$h=-5834496+77784=-5756712\text{ (not a solution)}$$

None of the evaluated heights are equal to zero for the selected $x$ values in the given table (12, 24, 36, or 48 feet). To precisely determine which $x$ value is closest to the x-intercept, we would typically need to solve the quadratic equation using numerical methods or a calculator.

However, based on the problem's structure, it’s implied that one of the table values is meant to be a solution when closely evaluated. In this context, the horizontal distance at which the firework reaches the ground is often at the maximum height, which can occur at the vertex of the parabola, but requires additional calculation methods.

Final Note:

If you are looking for a specific answer among the choices, please provide more context, or if I've made a computational error, clarify that with more detailed values. According to pure evaluation, no values from the table correspond to $h=0$. Therefore, if I were to select an answer from the list without additional information, it could be misleading, as none equally present the height as zero. Would you like to calculate the vertex point $x=-\frac{b}{2a}$ for a better estimate?

Answer 4

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

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
False
No, 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
54=0
True
Yes, 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
False
No, 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

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

Answer 5

To determine whether $x=-3$ is a solution to the quadratic equation

$$-3x^2-9x=0,$$

we need to substitute $x=-3$ into the equation and see if it results in a true statement (i.e., if both sides of the equation are equal).

Step-by-step substitution:

1. Substitute $x=-3$ into the equation: $$-3(-3{)}^2-9(-3)=0.$$

2. Calculate $(-3{)}^2$: $$=-3(9)+27.$$

3. Simplify the expression: $$-27+27=0.$$ $$0=0.$$

This confirms that substituting $x=-3$ into the equation yields a true statement.

Conclusion:

Based on this evaluation, the correct explanation is:

Yes, $x=-3$ is a solution because substituting it back into the equation results in the following:

$$\text{x Substituted Evaluate True Statement?}-3-3(-3{)}^2-9(-3)=00=0\text{(True)}$$

So the best response is the last one: 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 , \text{True}

Answer 6

Show how many solutions are in the solution set for the equation 3x2=21
.(1 point)
Responses

zero
zero

two
two

one
one

infinitely many
infinitely many

Answer 7

To determine how many solutions are in the solution set for the equation

$$3x^2=21,$$

we can first solve for $x$.

1. Divide both sides by 3:

$$x^2=\frac{21}{3}=7.$$

2. Take the square root of both sides:

$$x=\pm \sqrt{7}.$$

This gives us two solutions:

$$x=\sqrt{7}\text{and}x=-\sqrt{7}.$$

Conclusion:

Since there are two distinct solutions ($\sqrt{7}$ and $-\sqrt{7}$), the number of solutions in the solution set for the equation $3x^2=21$ is:

Two.

Answer 8

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

one
one

zero

Answer 9

To determine how many solutions are in the solution set for the equation

$$2(x-5{)}^2+2=20,$$

we can follow these steps:

Step 1: Simplify the equation

First, we will subtract 2 from both sides:

$$2(x-5{)}^2=20-2$$ $$2(x-5{)}^2=18.$$

Step 2: Divide both sides by 2

Now, we will divide both sides by 2:

$$(x-5{)}^2=\frac{18}{2}$$ $$(x-5{)}^2=9.$$

Step 3: Take the square root of both sides

Next, we take the square root of both sides:

$$x-5=\pm 3.$$

Step 4: Solve for $x$

Now we can solve for $x$:

1. For the positive case: $$x-5=3⟹x=3+5⟹x=8.$$

2. For the negative case: $$x-5=-3⟹x=-3+5⟹x=2.$$

Conclusion

We have found two solutions: $x=8$ and $x=2$.

Thus, the number of solutions in the solution set for the equation $2(x-5{)}^2+2=20$ is:

Two.