Completing the square with fractional coefficients

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Complete the square, making the vertex coordinates explicit:

• (a) $y=3+3x-5x^2$
• (b) $y=\dfrac{2x^2}{3}-x+1$

Solution

(a) $y=-5\left(x^2-\tfrac{3}{5}x\right)+3=-5\left(x-\tfrac{3}{10}\right)^2+\tfrac{69}{20}$. Vertex $V\left(\tfrac{3}{10};\tfrac{69}{20}\right)=(0{,}3;\,3{,}45)$.

(b) $y=\tfrac23\left(x^2-\tfrac32 x\right)+1=\tfrac23\left(x-\tfrac34\right)^2+\tfrac58$. Vertex $V\left(\tfrac34;\tfrac58\right)=(0{,}75;\,0{,}625)$.

$\boxed{V_a\left(\tfrac{3}{10};\tfrac{69}{20}\right),\qquad V_b\left(\tfrac34;\tfrac58\right)}$