**Solving Quadratic Equations by Factoring Using Box Method :**

In this section, you will learn how to solve quadratic equations by factoring using box method.

**Step 1 :**

Draw a box, split it into four parts.

Write the first and last term in the first and last box respectively.

**Step 2 :**

We have to multiply the coefficient of x^{2} term and constant term.

Now, we have to decompose the value that we get in step 2, such that the product must be equal to the value in step 2 and simplified value must be equal to the middle term.

**Step 3 :**

Write those values in the empty boxes. Factor horizontally and vertically

**Step 4 :**

Write the horizontal and vertical terms as pairs. By equating each factor to zero, we will get the values of x.

**Example 1 :**

Solve the following quadratic equation :

x^{2} - 3x - 10 = 0

**Solution : **

From the box method, we find the factors.

The factors are (x + 2) and (x - 5).

Then,

(x + 2)(x - 5) = 0

x + 2 = 0 x = -2 |
x - 5 = 0 x = 5 |

So, the solutions is {-2, 5}.

**Example 2 :**

Solve the following quadratic equation :

2x^{2} + x - 6 = 0

**Solution : **

From the box method, we find the factors.

The factors are (2x - 3) and (x + 2).

Then,

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

2x - 3 = 0 2x = 3 x = 3/2 |
x + 2 = 0 x = -2 |

So, the solution is {3/2, -2}.

**Example 3 :**

Solve the following quadratic equation :

√2x^{2} + 7x + 5√2 = 0

**Solution : **

From the box method, we find the factors.

The factors are (x + √2) and (√2x + 5).

Then,

(x + √2)(√2x + 5) = 0

x + √2 = 0 x = -√2 |
√2x + 5 = 0 √2x = -5 x = -5/√2 |

So, the solution is {-√2, -5/√2}.

**Example 4 :**

Solve the following quadratic equation :

2x^{2} - x + (1/8) = 0

**Solution : **

In the above quadratic equation, multiply each side by 8.

16x^{2 }- 8 x + 1 = 0

From the box method, we find the factors.

The factors are (4x - 1) and (4x - 1).

Then,

(4x - 1)(4x - 1) = 0

4x - 1 = 0 4x = 1 x = 1/4 |
4x - 1 = 0 4x = 1 x = 1/4 |

So, the solutions {1/4, 1/4}.

**Example 5 :**

Solve the following quadratic equation :

100x^{2} - 20 x + 1 = 0

**Solution : **

From the box method, we find the factors.

The factors are (10x - 1) and (10x - 1).

Then,

(10x - 1)(10x - 1) = 0

10x - 1 = 0 10x = 1 x = 1/10 |
10x - 1 = 0 10x = 1 x = 1/10 |

So, the solution is {1/10, 1/10}.

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equation by factoring using box method.

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