Let $7x^2 + 5x = h.$ What value of $h$ will give us exactly one solution for $x$?

Try not to use non-standard notation such as " $ " ... Instead, use commas and/or put each quantity on its own, separate line.

7x^2 + 5x = h can be rewritten as 7x^2 + 5x - h = 0. Then a=7, b=5 and c = - h.

The "discriminant" is b^2 - 4ac. If the discrim. is zero, then the quadratic equation will have TWO (not just one) real, equal roots.

Form b^2-4ac and set it = to 0. Then solve this equation for h:

5^2 - 4 (7) (-h) = 0 becomes 25 + 28h = 0. Solving for h, h = - 25/28.

If the discriminant is larger than 0, then the quadratic equation will have TWO real, different roots. If smaller than 0, then the q. e. will have two complex roots.