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Answer 16 and 17

Step-by-step explanation:

16

GivenEquation ax² + bx + c = 0One of the roots is twice the otherTo findShow that 2b² = 9acSolution

Let the roots be p and q

The sum and product of the roots:

p + q = -b/apq = c/a

The sum of the roots, considering p = 2q,

p + q = 2q + q = 3q

And

3q = -b/a ⇒ q = -b/(3a)

The product of the roots, considering p = 2q:

pq = 2q(q) = 2q²

And

2q² = c/a

Substitute q with -b/(3a)

2(-b/(3a))² = c/a2b²/(9a²) = c/a2b² = 9a²(c/a)2b² = 9ac

17.

Given

Quadratic equation

ax² = bx + c = 0, with equal rootsTo findShow that b² = 4acSolution

Equal roots means the discriminant is zero:

D = √b² - 4ac√b² - 4ac = 0b² - 4ac = 0b² = 4ac