Determine the value(s) of m so that the quadratic equation mx2+6x = −m has no real roots. (i.e. no solution)
Let's arrange it into the normal appearance of a quadratic
mx^2 + 6x + m = 0
to have no real roots, the discriminant < 0
b^2 - 4ac < 0
36 - 4(m)(m) < 0
4m^2 > 36
m^2 > 9
± m > 3
m > 3 OR m < -3
adding m ... m x^2 + 6x + m = 0
the discriminant ... b^2 - 4 a c ... is negative for no real roots
in this case ... 36 - 4 m^2 < 0 ... - 4 m^2 < -36 ... 4 m^2 > 36 ... m^2 > 9
-3 > m ... or ... m > 3
To determine the value(s) of m that would result in no real roots for the given quadratic equation mx^2 + 6x = -m, we can use the discriminant.
The discriminant is the part of the quadratic formula that determines the nature of the roots of a quadratic equation. For any quadratic equation of the form ax^2 + bx + c = 0, the discriminant can be calculated using the formula:
D = b^2 - 4ac.
In our case, the quadratic equation is mx^2 + 6x = -m. Comparing this equation to the standard form ax^2 + bx + c = 0, we can see that a = m, b = 6, and c = m.
Substituting these values into the discriminant formula, we get:
D = 6^2 - 4(m)(-m)
= 36 + 4m^2
For the equation to have no real roots, the discriminant (D) must be negative. So we set the discriminant less than zero:
36 + 4m^2 < 0
Now, let's solve this inequality to determine the values of m:
36 + 4m^2 < 0
First, subtract 36 from both sides:
4m^2 < -36
Next, divide both sides by 4 (since it is positive):
m^2 < -9
Taking the square root of both sides (keeping in mind that we are dealing with the imaginary square root of a negative number), we get:
m < ±√(-9)
Simplifying further, we have:
m < ±3i
So, the values of m that result in no real roots for the quadratic equation mx^2 + 6x = -m are m < 3i and m > -3i, where i is the imaginary unit.