Can someone please help me with these equations.
2x(x+2)(x^2 - 2x + 4) = 0
and
Simplify (be sure to rationalize all denominators)
3/[SQRT(5) - 2]
I hope you take more care with your math than you do with your spelling.
I can only guess that you mean ALGEBRA!
yes sorry
once an equation is factored and equal to zero, it is very easy to solve.
simply set each factor equal to zero and solve
the last bracket needs the quadratic formula to solve and you will get two imaginary solutions.
Let me know how you made out.
for the second, multiply top and bottom by the conjugate which is (√5 - 2)
surely you were shown how to do that, right?
Sure! Let's start by solving the equation:
2x(x+2)(x^2 - 2x + 4) = 0
To find the solutions, we need to set each factor equal to zero.
First factor:
2x = 0
Divide both sides by 2:
x = 0
Second factor:
x+2 = 0
Subtract 2 from both sides:
x = -2
Third factor:
x^2 - 2x + 4 = 0
Since this is a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -2, and c = 4.
x = (-(-2) ± √((-2)^2 - 4(1)(4))) / (2(1))
x = (2 ± √(4 - 16)) / 2
x = (2 ± √(-12)) / 2
Since there is a negative number inside the square root, the solutions will involve complex numbers. Simplifying further, we have:
x = (2 ± √(12)i) / 2
x = (2 ± 2√(3)i) / 2
x = 1 ± √(3)i
So the solutions to the equation are:
x = 0, x = -2, x = 1 + √(3)i, x = 1 - √(3)i
Now let's simplify the expression:
3/[SQRT(5) - 2]
To rationalize the denominator, we need to get rid of the square root. We can do this by multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of SQRT(5) - 2 is SQRT(5) + 2.
So we have:
3/[SQRT(5) - 2] * [SQRT(5) + 2]/[SQRT(5) + 2]
Using the distributive property, we can multiply:
(3 * (SQRT(5) + 2)) / [(SQRT(5) - 2) * (SQRT(5) + 2)]
Now, using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)), the denominator simplifies to:
[(SQRT(5))^2 - 2^2]
= (5 - 4)
= 1
Therefore, our expression simplifies to:
(3 * (SQRT(5) + 2)) / 1
= 3 * (SQRT(5) + 2)
= 3 * SQRT(5) + 6
So, the simplified expression is 3 * SQRT(5) + 6.
I hope that helps! Let me know if you have any further questions.