We are planned a picnic. The budget for this let ₹ x. If picnic increase 4 days, we decrease the daily expences. In general we think how many days planned for a picnic. We know this by quadratic equation also.
*& Find 'x' value by solving linear equation ax + b = c
ax = c − b
Here x power = 1, solution is unique.
*An equation of the form ax2 + bx + c = 0; a, b, c ∈ R & a ≠ 0, is called a quadratic equation in 'x'.
Polynomial of degree 2, then it has 2 roots.
Examples of quadratic equations:
1) x2 − 7x + 12 = 0
2) x2 + x - 30 = 0
3) 4x2 + 68x - 111 = 0
Examples for not a quadratic equations:
* p(x) is a degree of polynomial 2, it is in p(x) = 0 form then it is said to be quadratic equation.
* ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation ( p(x) in descending order of their degrees).
* y = ax2 + bx + c is called quadratic function.
Uses of Quadratic Functions:
First of all we see the example for this.
1) A train travels 360 km at a uniform speed. If the speed had been 10 km/hr more, it would have taken 3 hour less for the same journey. Find the speed of the train.
x2 + 10x = 1200
x2 + 10x − 1200 = 0
x2 + 40x − 30x − 1200 = 0
x(x + 40) − 30(x + 40) = 0
(x + 40)(x − 30) = 0
∴ x = 30 km/hr, x = −40 (Speed cannot negative)
∴ Actual speed of train = 30 km/hr.
By this quadratic functions/ quadratic equations have various uses.
1) Plane/Train/Bus speed increased/decreased, the distance it goes. We find the actual speed.
2) Find the profit/loss percentage, if we buy an article with profit/loss by using quadratic equation.
3) When the brakes are applied to a vehicle, the stopping distance is calculated.
4) When the rocket is fired upward, then the height of the rocket is defined by a quadratic function.
SOLUTION OF A QUADRATIC EQUATION BY FACTORISATION:
In general the zeroes of the quadratic polynomial ax2 + bx + c and the roots of he quadratic equation ax2 + bx + c = 0 are the same.
∴ No. of roots of the quadratic equation = 2
ax2 + bx + c be expressible as the product of two linear expressions.
ax2 + bx + c = (px + q)(rx + s) (p, q, r, s ∊ R & p ≠ 0, r ≠ 0)
ax2 + bx + c = 0 ⇒ (px + q)(rx + s) = 0
Example: Find the roots of the equation 6x2 + 11x + 3 = 0 by factorisation.
Sol: First of all we split the middle term of ax2 + bx + c = 0
We have to find two numbers p & q such that p + q = b & p ⨉ q = a ⨉ c
We have to find p + q = b = 11 & p ⨉ q = 6 ⨉ 3 = 18
List out all possible pairs of factors of 18
(1, 18), (2, 9), (3, 6), (−1, −18), (−2, −9), (−3, −6)
From the list it is clear that the pair (2, 9) will satisfy our condition
p + q = 11 & p ⨉ q = 18
∴ 11x can be written as 2x + 9x
6x2 + 11x + 3 = 6x2 + 2x + 9x + 3
= 2x[3x + 1] + 3[3x + 1]
= (2x + 3)(3x + 1)
∴ 6x2 + 11x + 3 = 0
⇒ (2x + 3)(3x + 1) = 0
2x + 3 = 0 3x + 1 = 0
SOLUTION OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE:
Some quadratic equations are not possible to solve by factorisation method. Then it is made a perfect square by adding or subtracting another term to it.
Ex: x2+ 6x + 6 = 0
x2 + 6x = −6
Adding both sides by '9'
x2 + 6x + 9 = −6 + 9
(x + 3)2 = 3
Square root both sides
STEPS FOR SOLVING QUADRATIC EQUATION ax2 + bx + c = 0
i) Divide each side by 'a'.
(iv) Write the LHS as a square and simplify the RHS.
(v) Solve it.
Ex: Find the roots of 5x2 − 7x − 6 = 0 by the method of completing the square.
Now, we consider the equation ax2 + bx + c = 0 where a, b, c ∈ R & a ≠ 0
ax2 + bx + c = 0; a, b, c ∈ R, a ≠ 0
ax2 + bx = − c
(This is also called as Shreedharacharya's rule)
Zero's are those points where value of polynomial becomes zero. Now we see the graph of this.
(i) If b2 − 4ac > 0, two roots are different. In such case if we draw the graph for this, curve of the quadratic equation cuts the x - axis at two distinct points.
(ii) If b2 − 4ac = 0 then two roots are equal and curve of the quadratic equation touching x - axis at one point.
(iii) If b2 − 4ac < 0, then there are no real roots. Roots are imaginary. In this case graph neither intersects nor touches the x - axis at all.
∴ b2 − 4ac is called the discriminant of the quadratic equation.
Ex: Find the nature of the roots of 4x2 + 3x + 5 = 0
Sol: b2 − 4ac = (3)2 − 4(4)(5)
= 9 − 80 = − 71 < 0
∴ Roots are not real.
