Answer as a fraction: 73/4
Answer in decimal form: 18.25
==================================================
Step-by-step explanation:
If alpha and beta are the roots of the quadratic ax^2+bx+c = 0, then we can say
- alpha+beta = -b/a
- alpha*beta = c/a
For more information refer to Vieta's Formulas. Focus on the quadratic case.
--------------------
Comparing 2x^2+5x-12 = 0 to ax^2+bx+c = 0, we have
Which leads us to
- alpha+beta = -b/a = -5/2 = -2.5
- alpha*beta = c/a = -12/2 = -6
Or in short
- alpha+beta = -2.5
- alpha*beta = -6
Let's square both sides of the first equation to get
alpha+beta = -2.5
(alpha+beta)^2 = (-2.5)^2
alpha^2 + 2*alpha*beta + beta^2 = 6.25
alpha^2 + 2*(-6) + beta^2 = 6.25
alpha^2 - 12 + beta^2 = 6.25
alpha^2 + beta^2 = 6.25 + 12
alpha^2 + beta^2 = 18.25 = 73/4
--------------------
Verification:
Use the quadratic formula to solve 2x^2+5x-12 = 0 to get the roots of x = -4 and x = 3/2 = 1.5
Those are the values of alpha and beta in either order.
alpha^2 + beta^2 = (-4)^2 + (1.5)^2 = 18.25 = 73/4
The answer is confirmed.