The equation ax^{2 }+ bx + c = 0 is the general or standard form of quadratic equation. For example: x^{2 }+5x + 12 = 0 and 5x^{2 }−4x − 16 = 0 are quadratic equations in the variable x. In competitive examinations, you must simplify a given equation before deciding whether it is quadratic or not.
In the equation above ‘a’ is termed the leading coefficient and ‘c’ is named the absolute term of f (x). The values of x that satisfy the quadratic equation are called the roots of the quadratic equation (denoted by;α,β). Quadratic equations are the basics of algebra and one must be familiar with them to be able to score well in the algebra section of any exam. Once the basics are clear, these equations become very easy to work with.
As per the definition, the standard form of a quadratic equation is ax^{2 }+ bx + c = 0, therefore the name quadratic comes as “quad” which means square because the variable gets squared(x^{2}) here.
For the equation: ax^{2 }+ bx + c = 0
These are also identified as polynomial equations of degree 2(due to the presence of x^{2}) in one variable. The important formula for quadratic equations in determining the roots is:
The quadratic equation in mathematics will always possess two roots and the nature of roots may be either real or imaginary depending upon the equation. Always remember that quadratic equations are seconddegree polynomial i.e if a given equation appears to be a thirddegree equation that is not quadratic. You will learn more about the above formula in the coming headings.
In the previous heading, we saw the formula to find the roots. Let us learn some other important formulas for quadratic equations for solving various types of questions in different examinations.
For a standard quadratic equation of the form ax^{2 }+ bx + c = 0.
The standard quadratic equations ax^{2 }+ bx + c = 0 being seconddegree equations in x hold two answers. These two answers for x are termed as the roots of the quadratic equations specified with the symbol (α, β). The question that comes here is how to find the roots of a quadratic equation.
One can find the roots of a quadratic equation using the formula;
Now that you know what are the roots of a given quadratic equation and how to find them. Let us step forward and learn about the nature of the root and how to find the nature of the roots of a quadratic equation.
The nature of the roots can be found by finding the roots using the formula. Another approach to determining nature is to find the value of b^{2}–4ac. Here, b^{2}–4ac is the part of the formula used to solve a quadratic equation. Let us understand how the nature of roots depends on this equation.
The general formula to obtain the roots is
If b^{2 }– 4ac = 0, then the nature of the roots α and β are real and equal.
If b^{2} − 4ac < 0, then the roots α and β are not real or the roots are imaginary.
If b^{2}− 4ac > 0 then the nature of the roots α and β are real and different/unequal.
If b^{2 }−4ac > 0 + a perfect square then the roots α and β are real, rational and unequal.
If b^{2 }− 4ac > 0 + not a perfect square, then the roots α and β are real, irrational and unequal.
To compose the standard form of a quadratic equation, the x^{2} term is penned first, followed by the x term, and eventually, the constant term is written. The value b^{2}–4ac denotes the discriminant of a quadratic equation, and is expressed by the symbol ‘D’.
For the quadratic equation ax^{2 }+ bx + c = 0 , the terms a,b and c specify the coefficient of x^{2}, x, and the constant term respectively. On the other hand, the roots of a quadratic equation are denoted by the symbols alpha (α), and beta (β). Now is there any relationship between the coefficient and roots of a quadratic equation? The answer would be yes.
The sum and product of roots of a quadratic equation, can be computed using these coefficients as shown:
The sum of the roots
The product of roots of quadratic equation
Also, if α and β, are the roots, then the quadratic equation can be formed using these roots. The equation is as follows: x^{2 }–(α + β)x + (αβ) = 0
Now that you know what a quadratic equation is with the definition and formula for solving such questions followed by information on the roots, their nature and roots of the quadratic equation formula. Let us now understand the different methods of solving quadratic equations.
Factoring quadratic equations is an approach where the equation ax^{2 }+ bx + c = 0is factorised as (x – ∝)(x – β) = 0 and then equation is solved to get x = ∝ or x = β. A standard form of quadratic equation in one variable can be solved by factorising the equation, i.e., by making factors out of the equation.
Consider the equation x^{2 }−7x + 12 = 0 , it can also be written as (x3)(x4)=0. Now this factorised equation can be very easily solved using a property of real numbers called the zeroproduct rule. We will cover the detailed procedure in the separate article.
Another approach for finding the roots is using the formula to find the roots of a quadratic equation. That is, solving the equation using the Sridharacharya formula. This can be a tricky method in terms of calculations but it is comparatively faster. This approach is generally used when the solution cannot be obtained using the factorisation method. Here the roots for the standard equation are obtained using the below formula:
Using the above formula you will get two roots of x one with a positive sign and the other with a negative sign as shown below:
Lastly, the solution can also be obtained by graphing quadratic equations. Consider the general form of a quadratic equation as a function of y, the equation becomes y = ax^{2 }+ bx + c
Next, substitute different values for x and obtain the respective values of y and plot the graph accordingly. The graph in general is a parabolashaped graph for the quadratic equation.
For the given two quadratic equations a_{1}x^{2 }+ b_{1}x + c_{1 }= 0 and a_{2}x^{2 }+ b_{2}x + c2 = 0. The condition for the two equations has a common root can be obtained as shown below.
The above equations are obtained using the determinant method.
Solving the equation we get:
Now squaring x and equating with x^{2} we get;
(a_{2}c_{1 }− a_{1}c_{2})^{2 }= (b_{1}c_{2 }− b_{2}c_{1})(a_{1}b_{2 }− a_{2}b_{1}) is required condition for the equation to have one common root.
However, if both the roots of quadratic equations, a_{1}x^{2 }+ b_{1}x + c_{1 }= 0 and a_{2}x^{2 }+ b_{2}x + c_{2 }= 0 are common then: a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2}
In the previous heading, we saw the condition for a standard form of a quadratic equation to have one and two common roots. For a single common root the condition is;
and for two common roots the condition is; a_{1}/a_{2 }= b_{1}/b_{2 }= c_{1}/c_{2} . Let us now understand how to solve such equations having common roots with an example.
Example 1: For the given quadratic equation 4x^{2 }+ 24x + 6 = 0 and 7x^{2 }– 6x + k = 0 what is the value of K for which the equations will have a common root?
Sol: (a_{2}c_{1 }− a_{1}c_{2})^{2 }= (b_{1}c_{2 }− b_{2}c_{1})(a_{1}b_{2 }− a_{2}b_{1}) is required condition for the equation to have one common root.
Form the equation, a_{1} = 4, b_{1} = 24, c_{1} = 6, a_{2} = 7, b_{2} = 6 and c_{2} = k
Substituting the values in the equation we get:
[(24k)−(−6×6)]×[4(−6)−(7×24)]=(7×6−4k)^{2}
[24k+36]×[−24−168]=(42−4k)^{2}
−4608k−6912=1764−336k+16k^{2}
16k^{2 }+ 4272k + 8676 = 0
Therefore, the values of k are 264.95, 2.04.
The general form of a quadratic function is represented by the equation y = ax^{2 }+ bx + c. As per the definition of the domain, it includes all the input values. Hence while determining the domain the concern of the graph is the horizontal axis, i.e. all the values of x. therefore, the domain is all real values.
Now coming towards the range: The range of a function is the collection of all possible output values. This implies that while calculating the range the focus area is the vertical axis, that is, the values on the y axis. The value of the range also depends on the opening of the parabola. To identify whether the parabola opens upwards or downward, check for the sign of the coefficient of x^{2}. If the sign of “a” is positive, then the parabola is concave upward and if “a” is negative, the parabola is concave downward.
Also, if the parabola is concave upward, the range will have all the real values that are greater than or equal to; Similarly, if the parabola is concave downward, the range will have all the real values less than or equal to;
Example 2: Find the range of the quadratic function given by the equation y = x^{2 }+ 8x + 12
Sol: On comparing the standard equation with the given one.
y = ax^{2 }+ bx + c
We get a = 1, b = 8 and c = 12
As the coefficient of x^{2} i.e. “a” is positive, the parabola is concave upward. Hence the range will have all the real values that are greater than or equal to
x = b / 2a
Substitute the values we get;
x = 8/2(12)
x = 8/24
x = 0.33
Substitute 0.33 for x in the given equation to obtain the ycoordinate at the vertex.
y=(−0.33)^{2 }+ 8(−0.33)+12
y = 0.1089 – 2.64+ 12
y = 9.46
Therefore, the ycoordinate of the vertex is 9.46.
As the parabola is concave upward, the range will have all the real values that are greater than or equal to 9.46.
The minimum, as well as the maximum value of the quadratic equation, depends on the nature of the graph, i.e. whether the graph opens upwards or downwards. When the graph is concave upwards i.e. a>0 then the expression holds a minimum value at x = b/2. On the other hand when the graph is concave downwards i.e. a<0 then the expression holds a maximum value at x = b/2a
In the previous heading, we use the two particular cases for the maximum and minimum values of the quadratic equations.
When the coefficient of x^{2} i.e. a > 0, the values lie in the ranges of [ f(b/2a), ∞)
When the coefficient of x^{2} i.e. a < 0, the values lie in the ranges of (∞, f(b/2a)]
Example 3: Locate the maximum or minimum value of the given quadratic equation −9(x–3)2+3
Sol: Given the equation; −9(x–3)^{2}+3
Solving the equation we get:
−9(x−3)^{2}+3 = −9(x^{2 }− 6x + 9) + 3 = −9x^{2 }+ 54x − 81 + 3
= −9x^{2 }+ 54x − 78
As, the coefficient of x^{2} is less than zero i.e. a < 0, the expression will hold a maximum value at x = b/2a
By substituting the values we get, x=3
Therefore, the maximum value of the quadratic equation−9(x – 3)^{2}+ 3 is 3.
A lot of daily life problems can be solved using quadratic equations. To solve any kind of word problem, first, you must translate the words into algebraic equations and then determine which method to apply to solve the equation so formed. Word problems are not specific to any one type. There can be innumerable types of word problems. Hence, there is no fixed technique or method to solve word problems. It’s all about logic and practice. However, if you keep the following points in mind, then it will be easy for you to solve any kind of word problem.
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1. What is the quadratic equation formula? 
2. How do you derive the quadratic equation formula? 
3. Can the quadratic equation formula be used for all types of quadratic equations? 
4. Are there any other methods to solve quadratic equations apart from using the quadratic formula? 
5. How can the quadratic equation formula be applied in reallife situations? 
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