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

Jul 6, 2022

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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 $$k = \color{brown}\boxed{16 \over 3}$$

Jul 6, 2022