Convert Standard Form of Quadratic Equation into Vertex Form
We know that the standard form of a quadratic equation is ax2 + bx + c = 0 and the vertex form is a(x – h)2 + k = 0 (where (h, k) is the vertex of the quadratic function.
Now we can easily convert the standard form into vertex form by comparing these two equations as,
ax2 + bx + c = a (x – h)2 + k
⇒ ax2 + bx + c = a (x2 – 2xh + h2) + k
⇒ ax2 + bx + c = ax2 – 2ahx + (ah2 + k)
Comparing coefficients of x on both sides,
b = -2ah
⇒ h = -b/2a … (1)
Comparing constants on both sides,
c = ah2 + k
⇒ c = a (-b/2a)2 + k (From (1))
⇒ c = b2/(4a) + k
⇒ k = c – (b2/4a)
⇒ k = (4ac – b2) / (4a)
Now the formulas h = -b/2a and k = (4ac – b2) /(4a) are used to convert the standard to vertex form.
Example of Converting Standard Form to Vertex Form
Consider the quadratic equation 3x2 – 6x + 4 = 0. Comparing this with ax2 + bx + c = 0, we get a = 3, b = -6, and c = 4. Now for vertex form, we found h and k
h = -b/2a
⇒ h = -(-6) / (2.3) = 1
⇒ k = (4ac – b2) / (4a)
⇒ k = (4.3.4 – (-6)2) / (4.3)
⇒ k = (48 – 36) / 12 = 1
Substituting a = 3, h = 1, and k = 1, the vertex form a(x – h)2 + k = 0 is,
3(x – 1)2 + 1 = 0
Standard Form of Quadratic Equation
Standard Form of the Quadratic Equation is ax2 + bx + c = 0, where a, b, and c are constants and x is a variable. Standard Form is a common way of representing any notation or equation. Quadratic equations can also be represented in other forms as,
- Vertex Form: a(x – h)2 + k = 0
- Intercept Form: a(x – p)(x – q) = 0
Standard Form of Quadratic Equation
In this article we will learn about the standard form of the quadratic equation, changing it into the standard form of the quadratic equation and others in detail.
Standard Form of Quadratic Equation