Write a quadratic equation with real coefficients if one of the solutions is -5 + 7i
complex roots always come in pairs, so the other root is -5 - 7i
sum of roots = (-5+7i) + (-5-7i) = -10
product of roots = (-5+7i)(-5-7i)
= 25 - 49i^2 = 25+49 = 74
equation:
x^2 + 10x + 74 = 0
Sure, here you go:
Well, since we want the quadratic equation to have real coefficients, the complex conjugate of -5 + 7i must also be a solution. The complex conjugate of -5 + 7i is -5 - 7i. So our quadratic equation becomes:
(x - (-5 + 7i))(x - (-5 - 7i)) = 0
Simplifying this, we get:
(x + 5 - 7i)(x + 5 + 7i) = 0
Now, let's expand it further:
(x + 5)(x + 5) + (x + 5)(7i) + (x + 5)(-7i) + (7i)(-7i) = 0
Combining like terms and simplifying some more, we have:
x^2 + 10x + 25 + 7ix + 5x + 35i - 5ix - 35i - 49 = 0
Now, let's group the real terms and the imaginary terms:
(x^2 + 10x + 25 - 49) + (7ix + 5x - 5ix + 35i - 35i) = 0
Combining these terms:
x^2 + 15x - 24 + 0i = 0
Finally, eliminating the imaginary term, we get our quadratic equation:
x^2 + 15x - 24 = 0
Voila! A quadratic equation with real coefficients and one of the solutions being -5 + 7i.
To write a quadratic equation with real coefficients, we know that if one solution is -5 + 7i, then the other solution must be the conjugate, which is -5 - 7i.
The quadratic equation can be written in the form:
(x - solution1)(x - solution2) = 0
Substituting the values, we have:
(x - (-5 + 7i))(x - (-5 - 7i)) = 0
Expanding the equation, we get:
(x + 5 - 7i)(x + 5 + 7i) = 0
Using the difference of squares, we have:
(x + 5)^2 - (7i)^2 = 0
Simplifying further:
(x + 5)^2 - 49i^2 = 0
Since i^2 is equal to -1, we can simplify it as:
(x + 5)^2 + 49 = 0
Therefore, the quadratic equation with real coefficients and one of the solutions as -5 + 7i is:
(x + 5)^2 + 49 = 0
To write a quadratic equation with real coefficients, we need to use the fact that complex solutions of a quadratic equation always come in conjugate pairs. Since one of the solutions is -5 + 7i, the other solution will be the conjugate of this complex number, which is -5 - 7i.
Now, let's start by using the solutions to form the quadratic equation. If we have a quadratic equation in the form ax^2 + bx + c = 0, where x represents the variable and a, b, and c are real coefficients, then we can use the following formula:
(x - solution1)(x - solution2) = 0
Substituting the values of the solutions, we get:
(x - (-5 + 7i))(x - (-5 - 7i)) = 0
Expanding this equation, we get:
(x + 5 - 7i)(x + 5 + 7i) = 0
Now, let's multiply the two binomials using the distributive property:
(x + 5)(x + 5) + (x + 5)(7i) - (x + 5)(7i) - (7i)(7i) = 0
Simplifying further, we have:
(x + 5)^2 - 49i^2 = 0
Remember that i^2 is equal to -1, so we can replace i^2 with -1:
(x + 5)^2 + 49 = 0
This is our quadratic equation with real coefficients:
(x + 5)^2 + 49 = 0