Question
Solve the following quadratic equation using the perfect square trinomial pattern: x2+x+0.25=0
Show your work.
Show your work.
Answers
Answered by
GPT 3.5
The given quadratic equation is: x^2 + x + 0.25 = 0
To solve this equation using the perfect square trinomial pattern, we need to rewrite the equation in the form of (x + a)^2 = 0, where 'a' is a constant.
Let's compare the given equation with the perfect square trinomial pattern:
x^2 + x + 0.25 = (x + a)^2
By comparing the terms, we can see that a = 0.5, since (0.5)^2 = 0.25.
So, we can rewrite the equation as:
x^2 + x + 0.25 = (x + 0.5)^2
Now, we can apply the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.
(x + 0.5)^2 = 0
Taking the square root of both sides:
x + 0.5 = 0
x = -0.5
Therefore, the solution to the quadratic equation x^2 + x + 0.25 = 0 using the perfect square trinomial pattern is x = -0.5.
To solve this equation using the perfect square trinomial pattern, we need to rewrite the equation in the form of (x + a)^2 = 0, where 'a' is a constant.
Let's compare the given equation with the perfect square trinomial pattern:
x^2 + x + 0.25 = (x + a)^2
By comparing the terms, we can see that a = 0.5, since (0.5)^2 = 0.25.
So, we can rewrite the equation as:
x^2 + x + 0.25 = (x + 0.5)^2
Now, we can apply the zero product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero.
(x + 0.5)^2 = 0
Taking the square root of both sides:
x + 0.5 = 0
x = -0.5
Therefore, the solution to the quadratic equation x^2 + x + 0.25 = 0 using the perfect square trinomial pattern is x = -0.5.
Answer
Ginger was asked to solve 25x^2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?
Show your work
Show your work
Answered by
GPT 3.5
To solve the quadratic equation 25x^2 - 196 = 0 using the difference of squares pattern, we rewrite it as:
(5x)^2 - 14^2 = 0
Now, we can apply the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b):
(5x + 14)(5x - 14) = 0
Now, set each factor equal to zero and solve for x:
5x + 14 = 0
5x = -14
x = -14/5
and
5x - 14 = 0
5x = 14
x = 14/5
Therefore, the correct solutions to the quadratic equation 25x^2 - 196 = 0 using the difference of squares pattern are x = -14/5 and x = 14/5.
(5x)^2 - 14^2 = 0
Now, we can apply the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b):
(5x + 14)(5x - 14) = 0
Now, set each factor equal to zero and solve for x:
5x + 14 = 0
5x = -14
x = -14/5
and
5x - 14 = 0
5x = 14
x = 14/5
Therefore, the correct solutions to the quadratic equation 25x^2 - 196 = 0 using the difference of squares pattern are x = -14/5 and x = 14/5.
Answer
Which of the following is a solution to the quadratic equation −x2+x=−10 ? Assume that the solution has been rounded to the nearest hundredth, if applicable. Show your work
Answered by
GPT 3.5
To find the solution to the quadratic equation -x^2 + x = -10, we rearrange the equation to the standard form:
x^2 - x + 10 = 0
We can solve this equation either by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -1, and c = 10.
x = (-(-1) ± √((-1)^2 - 4(1)(10))) / 2(1)
x = (1 ± √(1 - 40)) / 2
x = (1 ± √(-39)) / 2
Since the discriminant (√(b^2 - 4ac)) is negative, the equation has no real solutions. Therefore, there is no solution to the quadratic equation -x^2 + x = -10.
x^2 - x + 10 = 0
We can solve this equation either by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 1, b = -1, and c = 10.
x = (-(-1) ± √((-1)^2 - 4(1)(10))) / 2(1)
x = (1 ± √(1 - 40)) / 2
x = (1 ± √(-39)) / 2
Since the discriminant (√(b^2 - 4ac)) is negative, the equation has no real solutions. Therefore, there is no solution to the quadratic equation -x^2 + x = -10.
Answer
Which of the following coordinate pairs is a point on the parabola y=−4x^2−53x−56 ?
Show your work
Show your work
Answered by
GPT 3.5
To find the points on the parabola y = -4x^2 - 53x - 56, we can substitute different values of x into the equation to find the corresponding y-values. Let's evaluate the options using this method:
1) (0, -56)
When x = 0, y = -4(0)^2 - 53(0) - 56
y = 0 - 0 - 56
y = -56
2) (-2, -32)
When x = -2, y = -4(-2)^2 - 53(-2) - 56
y = -4(4) + 106 - 56
y = -16 + 106 - 56
y = 34
3) (1, -113)
When x = 1, y = -4(1)^2 - 53(1) - 56
y = -4 - 53 - 56
y = -113
4) (2, -180)
When x = 2, y = -4(2)^2 - 53(2) - 56
y = -4(4) - 106 - 56
y = -16 - 106 - 56
y = -180
Out of the given options, the coordinate pair (0, -56) is a point on the parabola y = -4x^2 - 53x - 56.
1) (0, -56)
When x = 0, y = -4(0)^2 - 53(0) - 56
y = 0 - 0 - 56
y = -56
2) (-2, -32)
When x = -2, y = -4(-2)^2 - 53(-2) - 56
y = -4(4) + 106 - 56
y = -16 + 106 - 56
y = 34
3) (1, -113)
When x = 1, y = -4(1)^2 - 53(1) - 56
y = -4 - 53 - 56
y = -113
4) (2, -180)
When x = 2, y = -4(2)^2 - 53(2) - 56
y = -4(4) - 106 - 56
y = -16 - 106 - 56
y = -180
Out of the given options, the coordinate pair (0, -56) is a point on the parabola y = -4x^2 - 53x - 56.
Answer
Which of the following coordinate pairs is a point on the parabola y=−4x2−53x−56 ?
(−4,−56) left parenthesis negative 4 comma negative 56 right parenthesis (−1,13) left parenthesis neagtive 1 comma 13 right parenthesis (−4,220) left parenthesis negative 4 comma 220 right parenthesis (−1,−7)
Show your work
(−4,−56) left parenthesis negative 4 comma negative 56 right parenthesis (−1,13) left parenthesis neagtive 1 comma 13 right parenthesis (−4,220) left parenthesis negative 4 comma 220 right parenthesis (−1,−7)
Show your work
Answered by
GPT 3.5
To determine which of the given coordinate pairs is a point on the parabola y = -4x^2 - 53x - 56, we need to substitute the x-value into the equation and check if the resulting y-value matches the given y-value. Let's evaluate each option:
1) (-4, -56)
When x = -4, y = -4(-4)^2 - 53(-4) - 56
y = -4(16) + 212 - 56
y = -64 + 212 - 56
y = 92
The calculated y-value is 92, which does not match the given y-value of -56. Therefore, (-4, -56) is not a point on the parabola.
2) (-1, 13)
When x = -1, y = -4(-1)^2 - 53(-1) - 56
y = -4(1) + 53 - 56
y = -4 + 53 - 56
y = -7
The calculated y-value is -7, which matches the given y-value of 13. Therefore, (-1, 13) is a point on the parabola.
3) (-4, 220)
When x = -4, y = -4(-4)^2 - 53(-4) - 56
y = -4(16) + 212 - 56
y = -64 + 212 - 56
y = 92
The calculated y-value is 92, which does not match the given y-value of 220. Therefore, (-4, 220) is not a point on the parabola.
4) (-1, -7)
When x = -1, y = -4(-1)^2 - 53(-1) - 56
y = -4(1) + 53 - 56
y = -4 + 53 - 56
y = -7
The calculated y-value is -7, which matches the given y-value of -7. Therefore, (-1, -7) is a point on the parabola.
In conclusion, the coordinate pairs that are points on the parabola y = -4x^2 - 53x - 56 are (-1, 13) and (-1, -7).
1) (-4, -56)
When x = -4, y = -4(-4)^2 - 53(-4) - 56
y = -4(16) + 212 - 56
y = -64 + 212 - 56
y = 92
The calculated y-value is 92, which does not match the given y-value of -56. Therefore, (-4, -56) is not a point on the parabola.
2) (-1, 13)
When x = -1, y = -4(-1)^2 - 53(-1) - 56
y = -4(1) + 53 - 56
y = -4 + 53 - 56
y = -7
The calculated y-value is -7, which matches the given y-value of 13. Therefore, (-1, 13) is a point on the parabola.
3) (-4, 220)
When x = -4, y = -4(-4)^2 - 53(-4) - 56
y = -4(16) + 212 - 56
y = -64 + 212 - 56
y = 92
The calculated y-value is 92, which does not match the given y-value of 220. Therefore, (-4, 220) is not a point on the parabola.
4) (-1, -7)
When x = -1, y = -4(-1)^2 - 53(-1) - 56
y = -4(1) + 53 - 56
y = -4 + 53 - 56
y = -7
The calculated y-value is -7, which matches the given y-value of -7. Therefore, (-1, -7) is a point on the parabola.
In conclusion, the coordinate pairs that are points on the parabola y = -4x^2 - 53x - 56 are (-1, 13) and (-1, -7).