Suppose that 2x^2 - 5x + k = -x^2 + 3x is a quadratic equation with one solution for x. Express k as a common fraction.

Guest Jul 6, 2022

#1**0 **

Simplify to \(3x^2 - 8x + k = 0\)

If the quadratic has exactly 1 solution, then it can be expressed as \(a(x-h)^2 \), where h is the x-coordinate of the vertex.

We know that \(a = 3\), because it is the leading coefficient of the \(x^2\) term.

Also, note that \(2h \times 3 = 8\), meaning \(h = {4 \over 3}\).

This means we have \(3(x-{4 \over 3})^2\), which expands to \(3x^2 - 8x + {16 \over 3}\), meaning \(k = \color{brown}\boxed{16 \over 3}\)

BuilderBoi Jul 6, 2022