5x^2-10x+d=0
Determine an integer value for d such that the equation has rational, non zero roots
Please show me the steps and how to answer this question :) thanks:)
depends on the value of the discriminant D
where D = b^2 - 4ac
remember that if
D > 0 we have 2 distinct real roots
D = 0 we have 1 real root
D <0 we have no real roots
So take all 3 cases , I will do one them, you do the rest.
D > 0
10^2 - 4(5)d > 0
-20d > -100
d < 5
one such value of d is -20
Wolfram shows 2 real roots
http://www.wolframalpha.com/input/?i=plot+y+%3D5x%5E2-10x+-+20
Oh ok , I was doing this exact thing and my answer was the same as yours but then when I checked my answers in my text book it gave me -20, that's why I was confused . Thank you so much:) really appreciate it:)
how do you divide
Well, let's put on our mathematical clown shoes and solve this equation!
The quadratic equation 5x^2 - 10x + d = 0 can be solve using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 5, b = -10, and c = d. We want the equation to have rational, non-zero roots, which means the discriminant (b^2 - 4ac) should be a perfect square.
Let's substitute the values into the formula:
x = (10 ± √((-10)^2 - 4 * 5 * d)) / (2 * 5)
x = (10 ± √(100 - 20d)) / 10
To make the square root term a perfect square, we need 100 - 20d to be a perfect square as well.
Using our clown intuition, we know that perfect squares can be any integer squared. So, let's try certain values of d.
For d = 0, we get 100 - 20(0) = 100, which is a perfect square.
For d = 1, we get 100 - 20(1) = 80, which is not a perfect square.
For d = 2, we get 100 - 20(2) = 60, which is not a perfect square.
Let's try another one, shall we?
For d = 5, we get 100 - 20(5) = 0, which is a perfect square!
There you have it, my friend! For d = 0 or d = 5, the equation 5x^2 - 10x + d = 0 will have rational, non-zero roots.
Always remember, in the wonderful world of mathematics, even clowns can find the perfect square!
To determine an integer value for d such that the equation has rational, non-zero roots, we need to use the discriminant. The discriminant formula is given as:
D = b^2 - 4ac
Where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
In this case, the quadratic equation is 5x^2 - 10x + d = 0. Comparing this equation to the standard form, we have a = 5, b = -10, and c = d.
To have rational, non-zero roots, the discriminant (D) must be a perfect square integer.
Therefore, we can substitute the values into the discriminant formula:
D = (-10)^2 - 4(5)(d)
D = 100 - 20d
Now, we want D to be a perfect square integer, which means there must exist an integer value for d that makes D a perfect square.
Let's set D = k^2, where k is an integer:
100 - 20d = k^2
Simplifying further:
20d = 100 - k^2
To find a value of d that makes this equation true, we need to find a value of k that results in the right-hand side being divisible by 20.
Here's a step-by-step process to find an integer value for d:
1. Start by choosing an integer value for k.
2. Square the chosen value of k (k^2).
3. Subtract k^2 from 100 to get (100 - k^2).
4. Check if (100 - k^2) is divisible by 20. If it is, proceed to the next step; otherwise, choose another value for k and repeat the process.
5. Divide (100 - k^2) by 20 to find the corresponding value of d.
By repeating steps 1 to 5, you can find an integer value for d that satisfies the condition of having rational, non-zero roots for the equation 5x^2 - 10x + d = 0.