Derive an equation whose coefficients are integers and which has roots of 1/2 and -7.
y = (2x-1)(x+7)
Why did the polynomial bring a calculator to the party?
Because it wanted to find its roots and coefficients on the go!
To derive an equation with roots of 1/2 and -7, we can start by subtracting each root from x:
For the root 1/2:
x - 1/2 = 0
For the root -7:
x + 7 = 0
Now, to get an equation with integer coefficients, we can multiply both equations together:
(x - 1/2) * (x + 7) = 0
Expanding the equation:
x^2 + 7x - (1/2)x - 7/2 = 0
Multiplying through by 2 to eliminate fractions:
2x^2 + 14x - x - 7 = 0
Simplifying further:
2x^2 + 13x - 7 = 0
And there you have it! The derived equation with integer coefficients, having roots of 1/2 and -7, is:
2x^2 + 13x - 7 = 0
To derive an equation with roots 1/2 and -7, we can use the fact that if a number, say "a," is a root of a polynomial equation, then (x-a) will be a factor of the equation.
In this case, we have two roots: 1/2 and -7. So, the equation can be written as:
(x - 1/2)(x + 7) = 0
Now, let's expand this equation to get an equation with integer coefficients:
(x - 1/2)(x + 7) = 0
x^2 + 7x - 1/2x - 7/2 = 0
(x^2 + 13/2x - 7/2) = 0
Multiplying the equation by 2 to eliminate the fractions:
2(x^2 + 13/2x - 7/2) = 0
2x^2 + 13x - 7 = 0
Therefore, the equation with integer coefficients and roots 1/2 and -7 is 2x^2 + 13x - 7 = 0.
To derive an equation with integer coefficients that has roots of 1/2 and -7, we need to find the expression for a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are integers.
First, let's use the fact that the roots of a quadratic equation are given by the solutions to the equation when it is set equal to zero.
So, for a root of 1/2, we can write an equation as follows:
(x - 1/2) = 0
Multiplying both sides by 2 to eliminate the fraction results in:
2(x - 1/2) = 0
2x - 1 = 0
2x = 1
x = 1/2
Similarly, for a root of -7, we can write an equation as follows:
(x + 7) = 0
Expanding it gives:
x + 7 = 0
x = -7
Now that we have the two roots, 1/2 and -7, we can express the quadratic equation as:
(x - 1/2)(x + 7) = 0
Next, let's expand this equation by multiplying the linear factors to obtain a quadratic equation:
(x - 1/2)(x + 7) = 0
(x * x) + (x * 7) + (-1/2 * x) + (-1/2 * 7) = 0
x^2 + 7x - (1/2)x - (7/2) = 0
To get rid of the fractional coefficients, we can multiply the entire equation by 2:
2(x^2 + 7x - (1/2)x - (7/2)) = 0
2x^2 + 14x - x - 7 = 0
2x^2 + 13x - 7 = 0
Therefore, the derived equation with integer coefficients whose roots are 1/2 and -7 is:
2x^2 + 13x - 7 = 0