Before knowing the standard form of a quadratic equation, lets get familiarized with quadratic equations
We have learnt about polynomials. Polynomials having degree two are quadratic polynomials.
It is of the form \(ax^2 + bx + c\), where \(a ≠ 0\).
When quadratic polynomial is equated to 0, it is called as quadratic equation.
\(ax^2 + bx + c\) = \(0\), where \(a ≠ 0\) is the standard form of quadratic equation.
Now, we will see the use of quadratic equation in real life
Consider a land having shape of rectangle. Length of \(l\) and is \(10~ meters\) more than breadth. What can be the length and breadth if area of the land is \(600 ~m^2\)?
Here, it is told that length is \(10~ meters\) more than breadth. If we consider breadth as a variable \(x\), then we can write the length in terms of \(x\).
Length of the land = \(x + 10\)
Area of the land = \(length~×~breadth\) = \((x+10)(x)\) which is equal to \(600\).
\((x + 10)(x)\) = \(600\)
\(x^2 + 10x\) = \(600\)
\(x^2 + 10x – 600\) = \(0\) —(1)
This is of the form \(ax^2 + bx + c\) = \(0\) —(2);
Comparing the equations (1) and (2) gives \(a\) = \(1\), \(b\) = \(10\), \(C\) = \(-600\)
- Equations \(2x^2 – 5 + 4x\) = \(0\), \(6 – x – 4x^2\) = \(0\), \(x + 4 + 2x^2\) = \(0\) are quadratic equations.
Therefore, any equation of the form \(P(x)\) = \(0\) where \(P(x)\) is a second degree polynomial is called quadratic equation. To get quadratic equation in standard form, we write the terms of the \(P(x)\) in descending order of their degrees.
i.e. standard form of the quadratic equation \(6 – x – 4x^2\) = \(0\) is \(4x^2 + x – 6\) = \(0\).
Similarly, \(x + 4 + 2x^2\) = \(0\) can be written as \(2x^2 + x + 4\) = \(0\) in the standard form.
There are many situations in real world, which can be represented mathematically by using quadratic equation.
For example; Arjun and Pratik had equal amount of cash in their wallet. In a tea shop, Arjun spent Rs. 40 and Pratik Rs. 50. Product of their remaining cash in the wallet is Rs. 1200. This can be represented mathematically as,
Let \(x\) be the initial amount they had in wallet.
After they spent cash in tea shop, Arjun and Pratik will have Rs. \((x – 40)\) Rs. \((x – 50)\) respectively remain in the wallet.
Since, product of remaining cash is equal to Rs. 1200. It can be written as,
\((x – 40)(x – 50)\) = \(1200\)
\(x^2 – 40x – 50x + 2000\) = \(1200\)
\(x^2 – 90x + 800\) = \(0\)
This is mathematical form of above situation.
Example: Verify whether following equations are quadratic equations or not.
- \(x^3 + x^2 + 1\) = \((x + 1)^3\)
\(x^3 + x^2 + 1\) = \(x^3 + 3x^2 + 3x + 1\)
\(2x^2 + 3x\) = \(0\)
which is of the form \(ax^2 + bx + c\) = \(0\), \(a ≠ 0\); where \(a\) = \(2\), \(b\) = \(3\) and \(c\) = \(0\)
Therefore, it is a quadratic equation.
- \(x^2 + 5x + 4\) = \((x + 2)^2\)
\(x^2 + 5x + 4\) = \(x^2 + 4x + 4\)
\(x\) = \(0\)
Which is not in the form \(ax^2 + bx + c\) = \(0\), a≠0.
Therefore it is not a quadratic equation.
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