To complete the square for the quadratic function [tex]\( y = 5x^2 + 15x + 4 \)[/tex], we will follow these steps:
Step 1: Factor out the coefficient of [tex]\( x^2 \)[/tex] from the terms involving [tex]\( x \)[/tex].
The coefficient of [tex]\( x^2 \)[/tex] is 5. Factoring 5 out from the terms involving [tex]\( x \)[/tex]:
[tex]\[ y = 5(x^2 + \frac{15}{5}x) + 4 \][/tex]
[tex]\[ y = 5(x^2 + 3x) + 4 \][/tex]
Step 2: Complete the square inside the parentheses.
Take the coefficient of [tex]\( x \)[/tex] (which is 3), divide it by 2, and then square it:
[tex]\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \][/tex]
Add and subtract [tex]\(\frac{9}{4}\)[/tex] inside the parentheses:
[tex]\[ y = 5(x^2 + 3x + \frac{9}{4} - \frac{9}{4}) + 4 \][/tex]
[tex]\[ y = 5\left((x + \frac{3}{2})^2 - \frac{9}{4}\right) + 4 \][/tex]
Step 3: Simplify the equation.
Distribute the 5 and simplify:
[tex]\[ y = 5(x + \frac{3}{2})^2 - 5 \cdot \frac{9}{4} + 4 \][/tex]
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{45}{4} + 4 \][/tex]
Convert 4 to a fraction with a denominator of 4:
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{45}{4} + \frac{16}{4} \][/tex]
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{45 - 16}{4} \][/tex]
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{29}{4} \][/tex]
Thus, the equation completed square form is:
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{29}{4} \][/tex]
Vertex Form:
The vertex form of a quadratic function is given by [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex. In this case:
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{29}{4} \][/tex]
From this form, we can identify the vertex:
[tex]\[ h = -\frac{3}{2}, \quad k = -\frac{29}{4} \][/tex]
Therefore, the vertex of the function is:
[tex]\[ \left(-\frac{3}{2}, -\frac{29}{4}\right) \][/tex]
In decimal form, the vertex is:
[tex]\[ \left(-1.5, -7.25\right) \][/tex]
So, the complete answer is:
Complete the square:
[tex]\[ y = 5(x + \frac{3}{2})^2 - \frac{29}{4} \][/tex]
Vertex:
[tex]\[ \left(-1.5, -7.25\right) \][/tex]
