The equation 2x² + px - 6 = 0 has equal roots. find the value(s) of p as a surd in its simplest form.

Hello from MrBillDoesMath!

There is no p value that answers the question.  

Discussion:

Consider the quadratic ax^2 + b^x + c = 0. The solution is

x =  ( -b +\- sqrt(b^2-4ac)) /2a

If b^2-4ac is NOT zero then there are two values of x that solve the equation. In b^2-4ac = 0, then both roots are equal, so there is only one solution.

Taking a = 2, b = p, and c = -6, in the quadratic formula:

b^2 - 4ac = 0 =>                   ( again, = 0 for a single root)

p^2 - 4(2)(-6) = 0 =>

p^2 + 48 = 0 =>                

which has no real solution as p^2 >= 0, so p^2 +48 >= 48 which means that p^2 + 48 =0 is not possible

Regards,  

MrB

(4,11), (6,14), (30,50)

Step-by-step explanation:

Since you are given two points, you can find the slope of the line and also the equation of the line. At first, it seemed like you could just use patterns and logic. One point on the line is (0,5) (this is the y-intercept and later we'll use this information) The next given point is (2,8) In order to get from (0,5) to (2,8) you must go up 3 and over (to the right) 2. Looking over the answers, we can see immediately that the next point "up 3 and over 2" is at (4, 11) which is one of the answer choices. But also, if you go once more "up 3 and over 2" you will land on (6,14) which is also an answer choice.

(5,10) CANNOT be an answer because in between the points (0,5), (2,8), (4,11), and (6,14) the line does not cross at an integer value. For example at x = 1 the y-value will be 6.5

However, we may as well write an equation for the line in order to check the points (30,50) and (40,60) The equation of the line using y=mx+b (this is called slope-intercept formula) is

y = (3/2)x + 5

We know this because that "up 3 and over 2" we were using is actually the slope, m, in the equation in front of the x, written as a fraction 3/2. The 5 is the y-intercept which we were given in the point (0,5)

Now if (30,50) or (40,60) are on the line then they will satisfy this equation.

y = (3/2)x + 5 when x = 30

y = (3/2)(30) + 5

y = 45 + 5

y = 50 so (30, 50) is also on this line

y = (3/2)x +5

y = (3/2)(40) + 5

y = 60 + 5

y = 65

So (40, 60) IS NOT on the line.

Alternatively you could use only slope formula to calculate the slope with the two given points (which you will find is 3/2. And then check each potential answer with one given point in the slope formula, keeping any point that also yields a slope of 3/2 and discarding any answers that do not give a slope of 3/2