1.) Simplify the expression.
[sqrt3 + 2i][sqrt3 - 2i]
2.) Solve using the square root property. (y - 4)2 = -18
3.) Rewrite the the function in vertex form. You may use completing the square or an alternate method.
f(x) = 5x2 - 10x + 3
4.) Use the Discriminant b2 - 4ac to determine the type and number of solutions. 6= 4x - 5x2
Thanks! <3
[sqrt3 + 2i][sqrt3 - 2i]
= 3 - 2i√3 + 2i√3 - 4i^2
= 3 - 4(-1) = 7
(y - 4)^2 = -18
y - 4 = ± √18
y = 4 ± 3√-2
= 4± 3√2 i
f(x) = 5x^2 - 10x + 3 , notice how we use exponents in this format
= 5(x^2 - 2x + 1 - 1) + 3
= 5 ( (x-1)^2 - 1) + 3
= 5(x-1)^2 - 5 - 3
= 5(x-1)^2 - 8
6= 4x - 5x^2
5x^2 - 4x + 6 = 0
b^2 - 4ac
= 16 - 4(5)(6) = -104
What do you think ?
For number 4 would the answer be "1 real"? Cause the options were 1 real, 2 real, or 2 complex.
No, since the disc I negative, you will have 2 complex roots
1.) To simplify the expression [sqrt3 + 2i][sqrt3 - 2i], we can use the difference of squares formula. The formula states that for any two complex numbers a + bi and a - bi, their product is equal to (a^2 - b^2).
In this case, we have sqrt3 + 2i and sqrt3 - 2i. If we let a = sqrt3 and b = 2, we can apply the formula.
Using the formula, we find that (sqrt3 + 2i)(sqrt3 - 2i) = (sqrt3)^2 - (2)^2 = 3 - 4 = -1.
Therefore, the simplified expression is -1.
2.) To solve the equation (y - 4)^2 = -18 using the square root property, we need to isolate the squared term on one side of the equation.
First, expand the squared term: y^2 - 8y + 16 = -18.
Next, move the constant term to the other side of the equation: y^2 - 8y + 34 = 0.
To apply the square root property, we need the equation to be in the form (x - a)^2 = b.
To achieve this, we complete the square by adding the square of half the coefficient of the y-term (which is -4) to both sides of the equation:
y^2 - 8y + (-4)^2 + 34 = (-4)^2.
This simplifies to: y^2 - 8y + 16 + 34 = 16.
Combining like terms gives: y^2 - 8y + 50 = 16.
Now we have the equation in the desired form. We can apply the square root property by taking the square root of both sides:
√(y^2 - 8y + 50) = √16.
y - 4 = ±√16.
Simplifying the square root of 16 gives: y - 4 = ±4.
Finally, we can solve for y by adding 4 to both sides of the equation:
y = 4 ± 4.
Therefore, the solutions are y = 8 and y = 0.
3.) To rewrite the function f(x) = 5x^2 - 10x + 3 in vertex form, we can use the method of completing the square.
First, factor out the common coefficient of 5 from the x^2 and x terms:
f(x) = 5(x^2 - 2x) + 3.
Next, we need to complete the square inside the parentheses. We take half the coefficient of the x-term and square it, and then add that value inside the parentheses while subtracting the same value multiplied by the common coefficient outside the parentheses.
Taking half of -2 gives -1, and squaring it gives 1. So we have:
f(x) = 5(x^2 - 2x + 1 - 1) + 3.
Simplifying inside the parentheses and factoring gives:
f(x) = 5((x - 1)^2 - 1) + 3.
Expanding the expression and simplifying:
f(x) = 5(x^2 - 2x + 1) - 5 + 3.
f(x) = 5x^2 - 10x + 5 - 5 + 3.
Finally, combining like terms gives the function in vertex form:
f(x) = 5(x - 1)^2 - 2.
Therefore, the function f(x) = 5x^2 - 10x + 3 rewritten in vertex form is f(x) = 5(x - 1)^2 - 2.
4.) To determine the type and number of solutions for 6 = 4x - 5x^2 using the discriminant (b^2 - 4ac), we need to rewrite the equation in the form ax^2 + bx + c = 0.
Rewriting the equation, we have: -5x^2 + 4x + 6 = 0.
Comparing this to the standard quadratic equation form, we can identify that a = -5, b = 4, and c = 6.
Now we can use the discriminant formula to calculate the discriminant:
Discriminant = b^2 - 4ac.
Substituting the values, we have:
Discriminant = (4)^2 - 4(-5)(6).
Simplifying this expression gives:
Discriminant = 16 + 120.
Discriminant = 136.
Since the discriminant is positive (136 > 0), this means the quadratic equation has two real solutions.
Therefore, the equation 6 = 4x - 5x^2 has two real solutions.