Last updated on July 5th, 2025
The quadratic equation is a second-degree polynomial, meaning that it is the highest degree exponent of two (x2). The word quadratic comes from quad, and it means square, as here the highest degree is two. ax2 + bx + c = 0 is the standard form of a quadratic equation.
The standard form of a quadratic equation is ax2 + bx + c = 0, where
x is the variable, a ≠ 0,
a, b, and c are real numbers.
Here, a is the coefficient of x2
b is the coefficient of x
And c is the constant
There are three forms to write the quadratic equation, including:
A second-degree polynomial equation is the quadratic equation, and its standard form is ax2 + bx + c = 0. The characteristics of the standard form of quadratic equations are:
The standard form of a quadratic function is written as:
f(x) = ax2 + bx + c = 0
Here, a, b, and c are the constant coefficients, and x is the variable. It is also known as the second-degree equation. In a quadratic function, the value of a ≠ 0, because if the value of a is 0, then the function will not be quadratic, as the highest degree in a quadratic is 2.
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) are the vertices of the quadratic function. To convert the equation from standard form to vertex form, we compare these two equations:
ax2 + bx + c = a(x - h)2 + k
Substituting the value of (x - h)2 in the equation, (x - h)2 = x2 - 2xh + h2
ax2 + bx + c = a (x2 - 2xh + hj) + k
ax2 + bx + c = ax2 -2axh + ah2 + k
Comparing the coefficients of x on both sides:
bx = -2axh
h = -bx/2ax
h = -b/2a, let’s consider this equation as (1)
Comparing the constants on both sides
c = ah2 + k
Substituting the value of h from (1)
c = a(-b/2a)² + k
c = a(b2/4a2) + k
c = (b2/4a) + k
c - (b2/4a) = k
k = c - (b2/4a)
Therefore, we can use the formulas h = -b/2a and k = c - (b2/4a) to convert a standard form of a quadratic equation into vertex form.
For example, convert 3x2 + 6x - 5 = 0 to vertex form
Here, a = 3
b = 6
c = -5
Given, equation is a(x - h)2 + k = 0
Finding the value of h and k:
h = -b/2a
h = -(6/2 × 3)
= -6/6
= -1
k = c - (b2/4a)
= -5 - (62/4 × 3)
= -5 - (36/12)
= -5 - 3
= -8
Substituting the value of h and k in: a(x - h)2 + k = 0
3(x - (-1))2 - 8 = 0
3(x + 1)2 -8
To convert vertex form to standard form, we simplify (x - h)2 = (x - h)(x - h). Let’s see how to convert with an example, converting 2(x + 3)² - 5
Here, a = 2
h = -3
k = -5
2(x + 3)² - 5 = 0
2(x + 3)(x + 3) - 5 = 0
2 (x2 + 6x + 9) -5 = 0
2x2 + 12x + 18 - 5 =0
2x2 + 12x + 13 = 0
How to Convert Standard Form of Quadratic Equation into Intercept Form?
The quadratic equation in intercept form is a(x - p)(x - q) = 0, where (p, 0) and (q, 0) are the x-intercepts. To convert a standard form to an intercept form, we first find the roots of the quadratic equation, as p and q are the roots of the quadratic equation. Let’s learn it with an example,
For example, converting the quadratic equation x2 - 7x + 12 = 0 into intercept form
We first find the root of the quadratic equation.
x2 - 7x + 12 = 0
Here, a = 1
b = -7
c = 12
To find the value of x we use quadratic equation:
x = (-b ± (√b2 - 4ac)2a
x = (-(-7) ± (√(-7)2 - (4 × 1 × 12)/2
x = (7 ± (√49 - 48))/2
x = (7 ± √1)/ 2
x = (7 ± 1)/2
So, x = (7 + 1)/2 ⇒ 8/2 = 4
x = (7 -1)/2 ⇒ 6/2 = 3
As x = 4 and x = 3
Therefore, p = 4 and q = 3
The intercept form of the quadratic equation is:
a(x - p)(x - q) = 0
Substituting the value of p and q:
1(x - 4)(x - 3) = 0
How to Convert Intercept Form to Standard Form?
To convert a quadratic equation in intercept form to standard form, we simply use the intercept form. In other words, by simplifying (x - p)(x - q) = 0.
For example, convert (2x + 3)(x -4) = 0 into standard form
(2x + 3)(x - 4) = 2x2 - 8x + 3x - 12 = 2x2 - 5x - 12
How to Represent Quadratic Functions in Standard Form in Graph?
The standard form of a quadratic function is f(x) = ax2 + bx + c, where a ≠ 0. The curve in the graph of a quadratic function is a parabola.
The width and slope of the parabola depend on the value of a. The vertex of the parabola is where the axis of symmetry crosses, and as it has the axis of symmetry, all parabolas are symmetric. The parabola opens upwards, and it opens downwards if a is negative.
In real-life, we use the standard form of a quadratic equation, where the relationships involve squared terms. The few applications of the standard form of quadratic equations
Students often find it hard to convert quadratic equations from one form to another. Here are some common mistakes and the ways to avoid them. Students can master the quartic equations by understanding these mistakes.
Convert x2 + 6x + 5 = 0 to vertex form
(x + 3)2 - 4 = 0
To convert a quadratic equation from standard to vertex form, we find the value of h and k using the formulas:
h = -b/2a
k = (4ac - b2)/4a
Here, a = 1
b = 6
c = 5
h = -b/2a
= -6/2×1 = -3
k = (4ac - b2)/(4a)
= ((4 × 1 × 5) - 62)/4 × 1
= 20 - 36/4
= -16/4
= -4
Here, h = -3 and k = -4
The standard form of vertex form is a(x - h)2 + k = 0
Substituting the value of h and k,
1(x - -3)2 + -4 = (x + 3)2 -4 = 0
x2 + 6x + 5 in vertex form is: (x + 3)2 -4 = 0
Convert 2(x + 1)2 - 5 = 0 to standard form
2(x - 2)(x + 1/2) = 0
To convert the standard form to intercept form, we first find the root of the quadratic equation, 2x2 - 3x - 2
Here, a = 2
b = -3
c = -2
x = (-b ± √b2 - 4ac)/2a
(-(-3) ± √(-3)2 - 4 × 2 × -2)/2×2
= (3 ± √9 - -16)/4
= (3 ± √25)/4
= (3 ± 5)/4
So, x = (3 + 5)/4 = 8/4 = 2
x = (3 - 5)/4 = -2/4 = -1/2
So, p = 2 and q = -1/2
So, the intercept form is:
a(x - p)(x - q) = 0
2(x - 2)(x - -1/2) = 0
2(x - 2)(x + 1/2) = 0
Convert 2x2 - 3x - 2 to intercept form
In intercept form 2x2 - 3x - 2 can be written as 2(x - 2)(x + 1/2) = 0
To convert a quadratic equation in vertex form to standard form, we simplify the equation
2(x + 1)2 - 5 = 0
Here, (x + 1)2 = x2 +2x + 1
So, the equation becomes:
2(x2 + 2x + 1) - 5 = 0
2x2 + 4x + 2 - 5 = 0
2x2 + 4x -3 = 0
Therefore, in intercept form 2x2 - 3x - 2 can be written as 2(x - 2)(x + 1/2) = 0
Convert x2 - 5x + 6 = 0 to intercept form
(x - 3)(x -2) = 0
To convert the quadratic equation from standard form to intercept form, first, we find the roots of the quadratic equation.
x = (-b ± √b2 - 4ac)/2a
Here, a = 1
b = -5
c = 6
Computing: √b2 - 4ac
√(-5)2 - 4 × 1 × 6
= √25 - 24
=√1
Substituting the value of √b2 - 4ac in (-b ± √b2 - 4ac)/2a
x = (-(-5) ± √1)/2
= (5 ± 1)/2
x = (5 + 1)/2 = 6/2 = 3
x = (5 - 1)/2 = 4/2 = 2
So, p = 3 and q = 2
Substituting the value of p and q in the equation:
a(x - p)(x - q) = 0
1(x - 3)(x - 2) = 0
(x - 3)(x -2) = 0
Convert 3(x - 1)(x + 5) = 0 to standard form
3x2 + 12 x - 15 = 0
To convert to standard form, we expand 3(x - 1)(x + 5) = 0
Expanding (x - 1)(x + 5):
(x - 1)(x + 5) = x2 + 5x - x - 5
= x2 + 4x - 5
Multiplying by 3: 3(x2 + 4x - 5)
3x2 + 12x - 15 = 0
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.