please help me solve these
1. 2√5/8 + 4√3/8 - 1/2√68
2. √4k-5 +1=8
(the radical sign goes over 4k-5)
3. (x+4)*2 /3 =8
4. Write the value of the discriminant of each equation. Then use it to decide how many different real-number roots the equation has.
x*2-6x+2=0
Answer to number 3:
2/3(x+4) = 8
2(x+4) = 24
2x+8 = 24
2x = 16
x = 8
Answer to number 3 is wrong. (didn't realise it was x+4 to the power of two thirds. Sorry):
Looks like Jenna is using * as the sign for exponents.
In #3 , no brackets were used around the 2/3, so it has to be read as
((x+4)^2)/3 = 8
then
(x+4)^2 = 24
x+4 = +/- sqrt(24)
x = -4 +/- 2sqrt(6)
Sure, let's solve these step by step:
1. To simplify the expression 2√5/8 + 4√3/8 - 1/2√68, we first need to simplify each term separately and then combine them.
For the first term, 2√5/8, you can simplify by dividing the numerator and denominator by the common factor of 2. This gives us √5 / 4.
Similarly, for the second term, 4√3/8, simplifying by dividing both numerator and denominator by 4 gives us √3 / 2.
For the third term, 1/2√68, you can simplify by dividing both the numerator and denominator by 2. This gives us 1/√17.
Now, we can combine the simplified terms: √5/4 + √3/2 - 1/√17.
To add or subtract square roots, we need to have the same radical. In this case, you can multiply the first term by 2/2 and the second term by 4/4 to obtain a common denominator of 4.
This gives us (2√5/4) + (2√3/4) - 1/√17.
Combining the terms, we have (2√5 + 2√3 - 1/√17) / 4.
That is the simplified form of the expression.
2. Let's solve the equation √(4k - 5) + 1 = 8.
First, isolate the square root term by subtracting 1 from both sides:
√(4k - 5) = 8 - 1
√(4k - 5) = 7
Next, square both sides to eliminate the square root:
(√(4k - 5))^2 = 7^2
4k - 5 = 49
Now, solve for k by adding 5 to both sides:
4k = 49 + 5
4k = 54
Finally, divide both sides by 4 to solve for k:
k = 54/4
k = 13.5
So the solution to the equation is k = 13.5.
3. Let's solve the equation (x + 4) * 2 / 3 = 8.
First, multiply both sides by 3 to eliminate the denominator:
3 * (x + 4) * 2 / 3 = 8 * 3
2(x + 4) = 24
Next, simplify the equation by distributing the 2 on the left side:
2x + 8 = 24
Then, isolate x by subtracting 8 from both sides:
2x = 24 - 8
2x = 16
Finally, divide both sides by 2 to solve for x:
x = 16 / 2
x = 8
So the solution to the equation is x = 8.
4. First, let's find the discriminant, which is the value inside the square root in the quadratic formula. For the equation x^2 - 6x + 2 = 0, the discriminant is given by b^2 - 4ac:
In this case, a = 1, b = -6, and c = 2. Plugging those values into the discriminant formula, we have:
b^2 - 4ac = (-6)^2 - 4(1)(2) = 36 - 8 = 28.
The discriminant is 28.
To determine how many different real-number roots the equation has, we can use the following rules:
- If the discriminant is positive (greater than 0), the equation has two distinct real roots.
- If the discriminant is zero, the equation has one real root.
- If the discriminant is negative, the equation has no real roots.
In this case, since the discriminant is positive (28), the equation x^2 - 6x + 2 = 0 has two different real-number roots.