Determine if the Following equality has real roots: (3*X^2) - (2*X) + 4 = (5*X^2) + (3*X) + 9, If the equation has real roots, calculate the roots for this equation.
From looking at this equality, we can see that this is a quadratic equation,This is because X2 is the highest order term we have in this equation:In order to find the roots of our equality, we need to rearrange the equation so that its in the form:0 = (aX2) + (bX) + cwhere a, b, and c are constants
Rearranging our equality:0 = (5X2) + (3X) + 9 - (3X2) + (2X) - 4 Collecting like terms:0 = (5-3)X2 + (3+2)X + (9-4)which simplifies to give us the equation:0 = (2X2) + (5X) +5
We can find the roots of a quadratic equation in 2 ways:FactorisationThe Quadratic Formula
When we try factorisation, we see that there is no way to factorise the equation, this means we have to use the quadratic formula:x = (-b +- sqrt(b2- 4ac))/(2a)
where 0 = (aX2) + (bX2) + c
from our rearranged equation, we have the following values for a, b, and c:a = 2 b = 5c = 5
substituting these values into our quadratic formula:
x = ( -5 +- sqrt(52 - 4(2)(5)))/(2*5)
simplifying we get that:x = (-5 +- sqrt(25 -40))/10which leads to:x = (-5 +- sqrt(-15))/10
Looking at the number inside the square root, we can see that this is -15.-15 is less than zero, from this, this means that there are NO REAL ROOTS FOR THIS EQUATION AS:sqrt(b2- 4ac) < 0
