Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Use the quadratic formula to solve the equation.
x2+5x−7=0
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2
+
5
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−
7
=
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go to part two for the rest of gaca math test 27.0831001 Adv Algebra Concpts & Conn A
To solve the equation x^2 + 5x - 7 = 0 using the quadratic formula, we first need to identify the values of a, b, and c in the general quadratic equation ax^2 + bx + c = 0.
In this case, a = 1, b = 5, and c = -7.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values of a, b, and c into the formula, we get:
x = (-(5) ± √((5)^2 - 4(1)(-7))) / (2(1))
Simplifying further:
x = (-5 ± √(25 + 28)) / 2
x = (-5 ± √(53)) / 2
Therefore, the solutions to the equation x^2 + 5x - 7 = 0 using the quadratic formula are:
x = (-5 + √(53)) / 2
and
x = (-5 - √(53)) / 2
To solve the quadratic equation using the quadratic formula, we need to first identify the coefficients of the terms in the equation.
For the given equation, x^2 + 5x - 7 = 0, we have the coefficients as follows:
a = 1 (coefficient of x^2)
b = 5 (coefficient of x)
c = -7 (constant term)
Now, we can substitute these values into the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Substituting the values, we get:
x = (-5 ± sqrt(5^2 - 4 * 1 * -7)) / (2 * 1)
Now let's simplify the formula step by step to find the solution:
x = (-5 ± sqrt(25 + 28)) / 2
x = (-5 ± sqrt(53)) / 2
Therefore, the solutions to the quadratic equation x^2 + 5x - 7 = 0 are:
x₁ = (-5 + sqrt(53)) / 2
x₂ = (-5 - sqrt(53)) / 2
These are the two possible values for x that satisfy the equation.
To solve the equation using the quadratic formula, we have to identify the values of a, b, and c in the standard form of a quadratic equation, which is ax^2 + bx + c = 0.
In this case, the equation is x^2 + 5x - 7 = 0. Comparing this with the standard form, we find:
a = 1
b = 5
c = -7
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Now, let's substitute the values of a, b, and c into the quadratic formula and solve for x:
x = (-5 ± √(5^2 - 4(1)(-7))) / (2(1))
x = (-5 ± √(25 + 28)) / 2
x = (-5 ± √53) / 2
Therefore, the solutions to the equation x^2 + 5x - 7 = 0 using the quadratic formula are:
x = (-5 + √53) / 2
x = (-5 - √53) / 2
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
What is the vertex form of the equation?
y=−x2+6x−7
To find the vertex form of the equation y = -x^2 + 6x - 7, we need to complete the square on the quadratic expression.
First, let's factor out the coefficient of x^2, which is -1:
y = -1(x^2 - 6x) - 7
Next, we need to find the value that completes the square in the expression x^2 - 6x. To do this, we take half of the coefficient of x and square it. Half of -6 is -3, and squaring -3 gives us 9. So we need to add 9 inside the parentheses to complete the square:
y = -1(x^2 - 6x + 9 - 9) - 7
Simplifying inside the parentheses:
y = -1((x - 3)^2 - 9) - 7
Expanding and simplifying further:
y = -(x - 3)^2 + 9 - 7
y = -(x - 3)^2 + 2
Thus, the vertex form of the equation y = -x^2 + 6x - 7 is y = -(x - 3)^2 + 2.
Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
What is the expression in factored form?
3x2+6x−24
To find the expression in factored form for 3x^2 + 6x - 24, we need to factor out any common factors and then factor the quadratic expression.
First, let's factor out the greatest common factor of the coefficients, which is 3:
3(x^2 + 2x - 8)
Now, we need to factor the quadratic expression (x^2 + 2x - 8). To do this, we need to find two numbers whose product is equal to the product of the coefficient of x^2 and the constant term (-8), and whose sum is equal to the coefficient of x (2).
The two numbers are 4 and -2:
3(x^2 + 2x - 8)
= 3(x + 4)(x - 2)
Therefore, the expression 3x^2 + 6x - 24 in factored form is 3(x + 4)(x - 2).
A student has simplified the following expression incorrectly.
Identify the mistake that was made and simplify the expression correctly. Show all necessary steps in the space provided.
(−3i)(−5i)
(
−
3
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)
(
−
5
�
)
15i
15
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The mistake in the student's simplification of (-3i)(-5i) is that they incorrectly multiplied the imaginary units.
To correctly simplify the expression (-3i)(-5i), we can treat the imaginary unit i as a variable and apply the rules of multiplication with variables:
(-3i)(-5i) = (-3)(-5)(i)(i)
Simplifying the real numbers, -3 and -5:
(-3)(-5)(i)(i) = 15(i)(i)
Now, we can simplify i^2, which is equal to -1:
15(i)(i) = 15(-1) = -15
Therefore, the correct simplification of (-3i)(-5i) is -15.