Table of Contents

**Quadratic Equation Class 10- Definition**

- A quadratic equation in the variable x is an equation of ax2+bx+c=0, where a,b, and c are real numbers, a≠0.
- For example, 2×2+x−300=0 is a quadratic equation.

**Quadratic Equation Class 10- Standard Form**

- Any equation of the form p(x)=0, where p(x) is a polynomial of degree 2, is a quadratic equation.
- But when we write the terms of p(x) in descending order of their degrees, we get the equation’s standard form.
- That is, ax2+bx+c=0, a≠0 is called the standard form of a quadratic equation.

**Quadratic Equation Class 10- Roots**

- A solution of the equation p(x)=ax2+bx+c=0, with a≠0, is called a root of the quadratic equation.
- A real number α is called a root of the quadratic equation ax2+bx+c=0,a≠0 if aα2+bα+c=0.
- It means x=α satisfies the quadratic equation or x=α is the root of the quadratic equation.
- The zeroes of the quadratic polynomial ax2+bx+c and the roots of the quadratic equation ax2+bx+c=0 are the same.

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**Quadratic Equation Class 10- Method Of Solving**

**1. Factorisation Method**

- Factorise the quadratic equation by splitting the middle term.
- After splitting the middle term, convert the equation into linear factors by taking common terms out.
- Then, on equating each factor to zero, the roots are determined.

For example:

⇒2×2−5x+3 (Split the middle term)

⇒2×2−2x−3x+3 (Take out common terms to determine linear factors)

⇒2x(x−1)−3(x−1)

⇒(x−1)(2x−3) (Equate to zero)\

⇒(x−1)(2x−3)=0

When (x−1)=0 , x=1

When (2x−3)=0 , x=32

So, the roots of 2×2−5x+3 are 1 and 32

**2. Method Of Completing The Square**

- The solution of a quadratic equation can be found by converting any quadratic equation to the perfect square of the form (x+a)2−b2=0.
- To convert quadratic equation x2+ax+b=0 to perfect square equate b, i.e., the constant term to the right side of the equal sign, then add a square of half of an, i.e., square of half of the coefficient of x on both sides.
- To convert the quadratic equation of form ax2+bx+c=0, a≠0 to perfect square, first divide the equation by an, i.e., the coefficient of x2, then follow the above-mentioned steps.

For example:

⇒x2+4x−5=0 (Equate constant term 5 to the right of the equal sign)

⇒x2+4x=5 (Add a square of half of 4 on both sides)

⇒x2+4x+(42)2=5+(42)2

⇒x2+4x+4=9

⇒(x+2)2=9

⇒(x+2)2−(3)2=0

It is of the form (x+a)2−b2=0

Now,

⇒(x+2)2−(3)2=0

⇒(x+2)2=9

⇒(x+2)=±3

⇒x=1

and x=−5

So, the roots of x2+4x−5=0are 1 and −5

**3. By using the quadratic formula**

The root of a quadratic equation ax2+bx+c=0 is given by the formula

x=−b±b2−4ac−−−−−−−√2a, where b2−4ac−−−−−−−√ is known as a discriminant.

If b2−4ac−−−−−−−√≥0, then only the root of a quadratic equation is given by

x=−b±b2−4ac−−−−−−−√2a

For example:

⇒x2+4x+3

By using the quadratic formula, we get

⇒x=−4±(4)2−4×1×3−−−−−−−−−−−−−√2×1

⇒x=−4±16−12−−−−−−√2

⇒x=−4±4–√2

⇒x=−4±22

⇒x=−4+22, x=−1

⇒x=−4−22,x=−3

So, the roots of x2+4x+3=0 are −1 and −3

**Quadratic Equation Class 10- Nature Of Roots Based On Discriminant**

- If b2−4ac−−−−−−−√=0, then the roots are real and equal
- If b2−4ac−−−−−−−√>0, then the roots are real and distinct.
- If b2−4ac−−−−−−−√<0then the roots are imaginary