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Complete the square to re-write the quadratic function in vertex form: y=x^(2)+3x-10 Answer: y=

Complete the square to re-write the quadratic function in vertex form:\newliney=x2+3x−10 y=x^{2}+3 x-10 \newlineAnswer: y= y=

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. Complete the square to re-write the quadratic function in vertex form:\newliney=x2+3x−10 y=x^{2}+3 x-10 \newlineAnswer: y= y=
  1. Identify xx-terms: To complete the square, we first need to focus on the xx-terms in the quadratic function y=x2+3x−10y = x^2 + 3x - 10. We will add and subtract the same value inside the equation to complete the square.
  2. Calculate half of coefficient: The coefficient of the xx-term is 33. To complete the square, we take half of this coefficient, square it, and add it to and subtract it from the equation. Half of 33 is 1.51.5, and 1.51.5 squared is 2.252.25. So we add and subtract 2.252.25 inside the equation.
  3. Add and subtract value: We rewrite the function as y=(x2+3x+2.25)−2.25−10y = (x^2 + 3x + 2.25) - 2.25 - 10. This allows us to form a perfect square trinomial with the first three terms.
  4. Rewrite the function: Now we factor the perfect square trinomial. The factored form of (x2+3x+2.25)(x^2 + 3x + 2.25) is (x+1.5)2(x + 1.5)^2 because (x+1.5)(x+1.5)(x + 1.5)(x + 1.5) gives us the original trinomial.
  5. Factor the perfect square trinomial: We then simplify the constants −2.25-2.25 and −10-10 by combining them to get −12.25-12.25. So the equation now reads y=(x+1.5)2−12.25y = (x + 1.5)^2 - 12.25.
  6. Simplify constants: The quadratic function is now in vertex form, which is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In our case, a=1a = 1, h=−1.5h = -1.5, and k=−12.25k = -12.25.

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