How do I solve this

Question

Find b and c so that y = 19x2

+ bx + c has vertex (7, 0).

Answer 1

The vertex form for a parabola is:

y = a(x-h)²+k where (h, k) is the vertex.

We know a = 19.

y = 19(x-7)²+0 must give the same result at y = 19x²+bx+c

So we have:

19[x²-14x+49) = 19x²+bx+c

This is a case for equating coefficients.

Distribute the 19 on the left side, noticing that the coefficient of x² will be 19 on both sides (expected).

The coefficient of x will be 19(-14) on the left and b on the right. So b = ?

Similarly for c.

Confirmation here:

desmos.com/calculator/nbuj6lobvu

Answer 2

If vertex is (7, 0) then

y = a(x - 7)² + 0, Vertext Form

y = ax² - 14x + 49

ax² - 14x + 49 = 19x² + bx + c

Can you answer now?