A quadratic equation contains a variable term with the exponent of 2 and no variable term with a higher power. A quadratic equation with one variable, x is an equation contains x

A quadratic equation is of the form

The general methods used to solve quadratic equations are:

The algebraic expression in the equation can be factored any of the
methods applied for factoring quadratic expressions like, GCF method,
splitting the middle term method or using binomial identities. The two
solutions of the quadratic equation are obtained by equating the factors
to zero using zero factor property and solving the linear equations so
formed. The steps involved in factoring a quadratic equation of the form ax^{2}as the highest power of x. A quadratic equation can have at most two solutions. A quadratic equation is also called a**second-degree equation.**A quadratic equation is of the form

**ax**, a $\neq$ 0. where a, b and c are real numbers.^{2}+ bx + c = 0The general methods used to solve quadratic equations are:

**Factoring method**- Taking square root method
- Completing the square
- Using Quadratic formula
- Graphical method

^{2}+ bx + c = 0 are,

**Steps for Solving Quadratic Equations:**

**Step 1:**Determine the values of a, b and c.

**Step 2:**Find the product "a * c".

**Step 3:**Choose any two numbers b

_{1}and b

_{2}such that b

_{1}b

_{2}= a * c and b

_{1}+ b

_{2}= b.

**Step 4:**Rewrite the equation as ax

^{2}+ (b

_{1})x + (b

_{2})x + c = 0

**Step 5:**Factor out the GCF pairwise.

**Repeat factoring using GCF.**

Step 6:

Step 6:

Before solving quadratic equations by using factorized method, let's see some examples for solving quadratic equations by GCF.

### Solved Examples

**Question 1:**Solve the quadratic equation 2x

^{2}- 4x = 0

**Solution:**

The Greatest common factor of the two terms 2x

Reversing the distribution in the expression 2x

2x(x - 2) = 0

Applying zero factor property,

either 2x = 0 or x - 2 = 0

=> x = 0 or x = 2

=> Solutions of the quadratic equation as

^{2}and 4x is 2x.Reversing the distribution in the expression 2x

^{2}- 4x by pulling 2x out , the equation is written in factored form as,2x(x - 2) = 0

Applying zero factor property,

either 2x = 0 or x - 2 = 0

=> x = 0 or x = 2

=> Solutions of the quadratic equation as

**x = 0 and x = 2****Question 2:**Solve the quadratic equation 3x

^{2}- 9x = 3x

**Solution:**

Given, quadratic equation 3x

=> 3x

=> 3x

The Greatest common factor of the two terms 3x

=> 3x

=> either 3x = 0 or x - 4 = 0

=> x = 0 or x = 4.

^{2}- 9x = 3x=> 3x

^{2}- 9x - 3x = 0=> 3x

^{2}-12x = 0The Greatest common factor of the two terms 3x

^{2}and 12x is 3x=> 3x

^{2}-12x = 3x(x - 4)=> either 3x = 0 or x - 4 = 0

=> x = 0 or x = 4.

**zero factor property**of a product states that if the product of two or more factors is zero, then at least one of the factors is equal to zero. This property is used in solving quadratic equations by factoring. A quadratic equation is a quadratic expression with an equal sign attached. The most commonly used technique for solving these equations is factoring. This method is also known an

**splitting the middle term method**.

The principle of zero product can be used in solving quadratic equations. Zero product states that if the product of two factors is zero, then at least one of the variable must be zero.

**If a . b = 0, then a = 0 or b = 0.**

### Solved Examples

**Question 1:**Solve the quadratic equation 2x

^{2}- 7x + 6 = 0

**Solution:**

Given quadratic equation 2x

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

Applying zero factor property to the factored equation,

either x - 2 = 0 or 2x - 3 = 0

=> x = 2 or x = $\frac{3}{2}$

Solving the binomials, the two solutions of the given equation are

^{2}- 7x + 6 = 0**a = 2 , b = -7 and c = 6**

Step 1:Step 1:

**Step 2:**ac = 2 x 6 =12**Step 3:**The numbers for split are -4 and -3 as their sum (-4) + (-3) = -7 = b and their product (-4)(-3) = 12 = ac.**Step 4:**The equation is rewritten as**2x**^{2}- 4x - 3x + 6 = 0**Step 5:**The terms are paired and factored using GCF method**2x(x - 2) - 3(x - 2) = 0**=> (x - 2)(2x - 3) = 0

**Step 6:**The common factor x - 2 is taken out and the distribution reversedApplying zero factor property to the factored equation,

either x - 2 = 0 or 2x - 3 = 0

=> x = 2 or x = $\frac{3}{2}$

Solving the binomials, the two solutions of the given equation are

**x = 2 and x = 1.5****Question 2:**Solve the quadratic equation x

^{2}- 5x + 6 = 0

**Solution:**

Given, quadratic equation x

Factor the left hand side of the equation,

=> x

= x(x - 2) - 3(x - 2)

= (x - 3)(x - 2)

=> x

Applying zero factor property to the factored equation,

either, x - 3 = 0 or x - 2 = 0

=> x = 3 or x = 2

=> Solutions of the given quadratic equation are

^{2}- 5x + 6 = 0Factor the left hand side of the equation,

=> x

^{2}- 5x + 6 = x^{2}- 2x - 3x + 6= x(x - 2) - 3(x - 2)

= (x - 3)(x - 2)

=> x

^{2}- 5x + 6 = (x - 3)(x - 2)Applying zero factor property to the factored equation,

either, x - 3 = 0 or x - 2 = 0

=> x = 3 or x = 2

=> Solutions of the given quadratic equation are

**x = 2, 3.**