Quadratic Equations Class 10 – Complete Notes, Formula, Examples & Practice Questions

1) What is a Quadratic Equation?

• A quadratic equation is an equation of degree 2.

The standard form is,

$$a{x}^2+bx+c=0$$

where

• $a \neq 0$
  
• $a, b,$ and $\mathrm{c\; are\; real\; numbers.}$

Examples

• $x^2-5x+6=0$
• $2x^2+3x-5=0$
  
• $5x^2-20=0$

2) Standard Form

Every quadratic equation can be written as

$$ax^2+bx+c=0$$

where

• a = coefficient of $x^2$
  
• b = coefficient of $x$
  
• c = constant term

Example:

$$3x^2-7x+2=0$$

Here,

• a = 3
• b = -7
• c = 2

3) Methods of Solving Quadratic Equations

There are three methods.

1. Factorization Method

The factorization method is the simplest way to solve a quadratic equation when it can be easily factored.

Steps:

• Write the equation in standard form.
• Factorize the quadratic expression.
• Set each factor equal to zero.
• Find the values of the variable.

Example:

$$x^2-5x+6=0$$

Factorizing,

$$(x-2)(x-3)=0$$

Therefore,

$$x=2,\;3$$

$$$$

$$$$Best for: Equations that factor into simple integers.

$$$$

$$$$

2. Completing the Square Method

• This method converts the quadratic equation into a perfect square, making it easier to solve.

Steps:

• Move the constant term to the other side.
• Add the square of half the coefficient of x to both sides.
• Write the left side as a perfect square.
• Take the square root of both sides.
• Solve for x.

Best for: Equations that cannot be factorized easily.

$$$$

Example

Solve:

$$x^2+6x+5=0$$

Step 1: Move the constant term.

$$x^2+6x=-5$$

Step 2: Half of 6 is 3, and $3^2=9$. Add 9 to both sides.

$$x^2+6x+9=-5+9$$

Step 3: Write the left side as a perfect square.

$$(x+3)^2=4$$

Step 4: Take the square root of both sides.

$$x+3=\pm2$$

Step 5: Find the values of $x$.

• $x+3=2 \Rightarrow x=-1$
  
• $x+3=-2 \Rightarrow x=-5$
  

Answer:

$$\boxed{x=-1,\,-5}$$

3. Quadratic Formula Method

• The quadratic formula is the most reliable method because it works for every quadratic equation, whether factorization is possible or not.

For the equation:

$$ax^2 + bx + c = 0$$

the solutions are:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Steps:

1. Identify the values of a, b, and c.
2. Substitute them into the quadratic formula.
3. Simplify the expression.
4. Find the two possible values of x.

Best for: All quadratic equations, especially when the equation has irrational or complex roots.

Example

Solve the equation:

$$2x^2+7x-4=0$$

Step 1: Compare with $ax^2+bx+c=0$

• $a=2$
  
• $b=7$
  
• $c=-4$
  

Step 2: Substitute the values into the quadratic formula.

$$x=\frac{-7\pm\sqrt{7^2-4(2)(-4)}}{2(2)}$$

Step 3: Simplify.

$$x=\frac{-7\pm\sqrt{49+32}}{4}$$​ $$x=\frac{-7\pm9}{4}$$

Step 4: Find the roots.

• $x=\dfrac{-7+9}{4}=\dfrac{2}{4}=\dfrac{1}{2}$
• $x=\dfrac{-7-9}{4}=\dfrac{-16}{4}=-4$
  

Answer:

$$\boxed{x=\frac{1}{2}\text{ or }x=-4}$$

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