# Quadratic Equations

## Quadratic Equations Synopsis

**Synopsis**

- An equation of the form ax
^{2}+bx +c = 0 is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.

**Roots of the quadratic equation **

- The value of x that satisfies an equation is called the roots of the equation.
- A real number a is said to be a solution/root of the quadratic equation ax
^{2}+ bx + c = 0 if a*a*^{2}+ b*a*+ c = 0. - A quadratic equation has
**at****most two roots**

**Solving Quadratic Equation by:**

- Splitting the middle term (or factorization) method

If ax^{2}+ bx + c = 0, a ≠ 0, can be reduced to the product of two linear factors, then the roots of the quadratic equation ax^{2}+ bx + c = 0 can be found by equating each factor to zero.

Steps involved in solving quadratic equationax^{2}+bx+c = 0(a ≠0) by splitting the middle term (or factorization) method:**Step 1:**Find the product ac.**Step 2:**Find the factors of ‘ac’ that add up to b, using the following criteria:

i.If ac > 0 and b > 0, then both the factors are positive.

ii.If ac > 0 and b < 0, then both the factors are negative.

iii.If ac < 0 and b > 0, then larger factor is positive and smaller factor is negative.

iv.If ac < 0 and b < 0, then larger factor is negative and smaller factor is positive.**Step 3:**Split the middle term into two parts using the factors obtained in the above step.**Step 4:**Factorize the quadratic equation obtained in the above step by grouping method. Two factors will be obtained.**Step 5:**Equate each of the linear factors to zero to get the value of x. **Completing the square method**

Any quadratic equation can be converted to the form (x + a)^{2}– b^{2}= 0 or (x – a)^{2}+ b^{2}= 0 by adding and subtracting the constant term. This method of finding the roots of quadratic equation is called the method of completing the square.

The steps involved in solving a quadratic equation by completing the square, are as follows:**Step 1:**Make the coefficient of x^{2}unity.**Step 2:**Express the coefficient of x in the form 2⨯ x ⨯ p.**Step 3:**Add and subtract the square of p.**Step 4:**Use the square identity (a + b)^{2}or (a – b)^{2}to obtain the quadratic equation in the required form (x + a)^{2}- b^{2}= 0 or (x – a)^{2}+ b^{2}= 0.**Step 5:**Take the constant term to the other side of the equation.**Step 6:**Take the square root on both the sides of the obtained equation to get the roots of the given quadratic

equation.**Quadratic formula**

The roots of a quadratic equation ax^{2}+ bx + c = 0 (a ≠ 0) can be calculated by using the quadratic**formula:**

If b^{2}– 4ac < 0, then equation does not have real roots.

**Discriminant of a quadratic equation**

For the quadratic equation ax^{2}+ bx + c = 0, a ¹ 0, the expression b^{2}– 4ac is known as discriminant.

**Nature of Roots**

Let f(x) = ax^{2}+ bx + c = 0 be the quadratic equation, the discriminant D = b^{2}– 4ac.**Relation between roots and coefficients:**

Let a, b be the roots of the quadratic equation ax^{2}+ bx + c = 0 then

**Note:** Quadratic equation can be rewritten as.x2-(α+β) x+ α.β =0

**Equations reducible to Quadratic Equations**

There are many equations which are not in the quadratic form but can be reduced to the quadratic form by simplifications.**Type I:**ax^{2}n + bxn + c = 0

Put xn = y

So, the equation reduces to ay^{2}+ by +c = 0

Now solve for y and hence for x.**Type II:**where a, b, c are constants

Make it az^{2}– cz + b = 0

Now, solve it for z.**Type III:**(x+a) (x=b) (x+c) (x+d)+ k = 0,where the sum of two of the quantities is equal to the sum of the other tw)o.

Example:

(x+1)(x+2)(x+3)(x+4) + 1= 0

As sum of 1 and 4 is equal to sum of 2 and 3

⟹ [(x+1)(x+4)][(x+2)(x+3)+1 =0

⟹ (x^{2}+ 5x +4 )(x^{2}+5x+6)+1=0

Take x^{2}+5x =y

⟹ (y+4)(y+6)+1=0

⟹ y^{2}+10y+24+1=0

⟹ y^{2}+10y +25 =0

⟹ (y+5)^{2}=0

⟹ y=-5

⟹ x^{2}+5x+5=0

**Formation of Equations with given roots**

Suppose the given roots are α and β

Therefore x = α and x =β

⟹ (x- α) = 0 and (x- β ) =0

The equation forms will be

(x- α)(x- β)=0

⟹ x^{2}–( α+ β)x+ α β=0

⟹ x^{2}-(sum of roots)x + Product of roots =0

**Applications of quadratic equations**- The applications of quadratic equation can be utilized in solving real life problems.
- Following points can be helpful in solving word problems:
- Every two digit number ‘xy’ where x is a ten’s place and y is a unit’s place can be expressed as. Xy= 10x +y.
- Downstream: It means that the boat is running in the direction of the stream

Upstream: It means that the boat is running in the opposite direction of the stream

Thus, if

Speed of boat in still water is x km/h

And the speed of stream is y km/h

Then the speed of boat downstream will be (x + y) km/h and in upstream it will be (x − y) km/h.

If a person takes x days to finish a work, then his one day's work

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