To convert a quadratic equation from standard form to vertex form, we need to complete the square. The standard form of the quadratic equation given is:
[tex]\[ y = 5x^2 + 20x + 14 \][/tex]
Here are the steps to convert it to vertex form:
1. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[ y = 5(x^2 + 4x) + 14 \][/tex]
2. Complete the square:
- Take the coefficient of [tex]\(x\)[/tex] (which is 4), divide it by 2, and square it. The result is [tex]\( \left(\frac{4}{2}\right)^2 = 4 \)[/tex].
- Add and subtract this square inside the parenthesis:
[tex]\[ y = 5\left(x^2 + 4x + 4 - 4\right) + 14 \][/tex]
3. Rewrite the expression inside the parenthesis as a square minus 4:
[tex]\[ y = 5\left((x + 2)^2 - 4\right) + 14 \][/tex]
4. Distribute 5 and simplify:
[tex]\[ y = 5(x + 2)^2 - 20 + 14 \][/tex]
[tex]\[ y = 5(x + 2)^2 - 6 \][/tex]
Thus, the vertex form of the equation is:
[tex]\[ y = 5(x + 2)^2 - 6 \][/tex]
Out of the given options, the correct one is:
D. [tex]\( y = 5(x+2)^2 - 6 \)[/tex]
