I have a take home test and I'm stuck on a couple. Anyone help please?
1. Find the vertex & axis of symmetry: -5(x+4)^2-4
2. Find the vertex & axis of symmetry: y=x^2-10x+3
3. Simplify: (2i+5)(3+4i)
4. Simplify: 3(21+2i)+(4-5i)
5. Simplify: (27x^2y^3)^-3/4
6. Simplify: 5�ã300x^4-2�ã243x^4
7. Simplify: 3�ã3(4�ã3+5�ã2)
8. Simplify: (2�ã3+�ã6)^2
9. Simplify: ^4�ã50 divided by ^4�ã2
10. 7 divided by ^3�ã4
11. 4 divided by �ã5-2
note: The a keys are supposed to be square root. I guess the sign doesn't show up here. Also, some are to the power of.
To answer number 3 use f.o.i.l and then wherever you get isquared turn to a negative 1
(2i+5)(3+4i)
6i + 8i^2 + 15 + 20i
6i +8(-1) + 15 + 20i
6i - 8 + 15 + 20i
26i + 7
the first two questions form the basics of this topic on quadratic equations.
If you cannot determine the vertex from
y = -5(x+4)^2-4
I think you are in deep trouble.
#3 is done for you, do #4 the same way
#5 (27x^2y^3)^-3/4 = 1/(27x^2y^3)^+3/4
any more "simplification" would only make it look more complicated
#6 I cannot tell where your square root ends.
Is it (5�√300)x^4-2�(√243)x^4
or
5�√(300x^4)-2�√(243)x^4) ?
if the first, then
=5*(10√3)x^4 - 2(9√3)x^4
= (32√3)x^4
#7,8, just expand them
#9 and #10, don't know what you mean by
^4√50, there is no base.
Of course, I'll be happy to help you with your test questions! I will provide step-by-step explanations for each problem so you can understand how to arrive at the answers.
1. Find the vertex & axis of symmetry: -5(x+4)^2-4
To find the vertex and axis of symmetry of a quadratic equation in the form y = a(x-h)^2 + k, where (h,k) represents the vertex, you can use the following steps:
- The x-coordinate of the vertex is given by the formula -h, so in this case, h = -4.
- The y-coordinate of the vertex is given by the value of k, so in this case, k = -4.
- Therefore, the vertex is (-4, -4).
- The axis of symmetry is a vertical line that passes through the vertex, so in this case, it is x = -4.
2. Find the vertex & axis of symmetry: y = x^2 - 10x + 3
Similarly, using the same steps:
- The x-coordinate of the vertex is given by the formula -h, where h is the coefficient of the x-term divided by 2. In this case, h = 10/2 = 5.
- The y-coordinate of the vertex is obtained by substituting the x-coordinate into the equation: y = (5)^2 - 10(5) + 3 = -22.
- Therefore, the vertex is (5, -22).
- The axis of symmetry is a vertical line that passes through the vertex, so in this case, it is x = 5.
3. Simplify: (2i+5)(3+4i)
To simplify this expression, you can use the distributive property of multiplication over addition:
- Multiply 2i by 3: 2i * 3 = 6i.
- Multiply 2i by 4i: 2i * 4i = 8i^2.
- Multiply 5 by 3: 5 * 3 = 15.
- Multiply 5 by 4i: 5 * 4i = 20i.
Since i^2 = -1, we can substitute it into the expression:
- 8i^2 becomes 8(-1) = -8.
Combining the terms, we have: 6i + 20i + 15 - 8 = 26i + 7.
Therefore, (2i+5)(3+4i) simplifies to 26i + 7.
4. Simplify: 3(21+2i)+(4-5i)
To simplify this expression, you follow the order of operations:
- Multiply 3 by each term inside the parentheses: 3 * 21 + 3 * 2i = 63 + 6i.
- Distribute 3 to the second parentheses: 3 * 4 + 3 * (-5i) = 12 - 15i.
- Combine like terms: (63 + 6i) + (12 - 15i) = 63 + 12 + (6i - 15i) = 75 - 9i.
Therefore, 3(21+2i)+(4-5i) simplifies to 75 - 9i.
5. Simplify: (27x^2y^3)^(-3/4)
To simplify this expression, you can apply the power of a power rule:
- Raise the base (27x^2y^3) to the power (-3/4).
- Separate the exponents: 27^(-3/4) * (x^2)^(-3/4) * (y^3)^(-3/4).
Simplify the exponents:
- For 27^(-3/4), the denominator of the fractional exponent gives the root, so it becomes 1/(27^(3/4)).
- For (x^2)^(-3/4), the exponent inside the parentheses means we take the fourth root followed by cubing it, resulting in 1/(x^(3/2)).
- For (y^3)^(-3/4), similar to the second term, it becomes 1/(y^(9/4)).
So, (27x^2y^3)^(-3/4) simplifies to [1/(27^(3/4)) * 1/(x^(3/2)) * 1/(y^(9/4))].
6. Simplify: 5√300x^4 - 2√243x^4
To simplify this expression, you can apply the rules for simplifying radicals:
- Simplify the numbers under the square root: √300 can be simplified to 2√75, and √243 can be simplified to 3√3.
- Combine like terms with the variable x^4: 5√75x^4 - 2√3x^4.
- Simplify further if possible.
7. Simplify: 3√3(4√3 + 5√2)
To simplify this expression, you can use the distributive property:
- Multiply 3√3 by each term inside the parentheses: 3√3 * 4√3 = 12√9 = 12 * 3 = 36 and 3√3 * 5√2 = 15√6.
- Therefore, the expression simplifies to 36√3 + 15√6.
8. Simplify: (√3 + √6)^2
To simplify this expression, you can use the FOIL method or the formula (a + b)^2 = a^2 + 2ab + b^2:
- Expand (√3 + √6)^2 using the FOIL method: (√3 + √6)(√3 + √6) = 3 + 2√3√6 + 6.
- Simplify further if possible.
9. Simplify: ^4√50 divided by ^4√2
To simplify this expression, you can use the quotient rule for radicals:
- Divide the roots: ^4√50 / ^4√2 = ^4√(50 / 2) = ^4√25.
- Simplify further if possible.
10. 7 divided by ^3√4
To simplify this expression, you can multiply the denominator by the cube root of the number:
- Multiply 7 by ^3√(4^2) = 7 * ^3√16.
- Simplify further if possible.
11. 4 divided by √5 - 2
To simplify this expression, you can use rationalization:
- Multiply the numerator and denominator by the conjugate of the denominator, which is √5 + 2. This will eliminate the square root in the denominator.
- Multiply 4 by √5 + 2: 4 * (√5 + 2).
- Multiply -(√5 - 2) by (√5 + 2): -(√5 - 2)(√5 + 2).
- Simplify further if possible.