The vertex form of a quadratic has the form y = m(x – s)2 + t for some m, s and t. This form is used to easily find the vertex of a quadratic which can be found at point (s, t).
To take a quadratic in the form y = ax2 + bx + c and factor it into y = m(x – s)2 + t, we must complete the square.
Completing the Square
We will use an example to demonstrate. Complete the square of y = 3x2 + 12x + 15
Step 1: Factor out the coefficient of x2 from the first two terms.
y = 3(x2 + 4x) + 15
Step 2: Calculate 0.25b2 where b is the coefficient of x.
0.25b2 = 0.25(4)2 = 4
Step 3: Add and subtract that value inside the brackets to keep the function the same.
y = 3(x2 + 4x + 4 – 4) + 15
Step 4: Expand the negative term.
y = 3(x2 + 4x + 4) + 3
Step 5: Factor the perfect square.
y = 3(x + 2)2 + 3
So our quadratic in vertex form is y = 3(x + 2)2 + 3 and the vertex can be found at (-2, 3).
Additionally we can see that the quadratic opens upwards because the value of both a and m are positive. Thus we can tell there are no real roots because the vertex is above the x-axis and the parabola opens upwards.
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This article was written for you by Jeremie, one of the tutors with Test Prep Academy.