**Solving Quadratic Equations Using the Quadratic Formula :**

In this section, how we solve quadratic equation using formula.

**How to use quadratic formula to solve quadratic equation ?**

(i) Compare the given quadratic equation with the general form of quadratic equation ax² + bx + c = 0

(ii) Here the coefficient of x² is a, coefficient of x is b and the constant term is c.

(iii) Then apply those values in the formula

**Example 1 :**

Solve using quadratic formula

a (x^{2} + 1) = x (a^{2} + 1)

**Solution :**

a (x^{2} + 1) = x (a^{2} + 1)

ax^{2} + (a^{2} + 1)x + a = 0

By comparing the given quadratic equation with general form of a quadratic equation,

ax^{2} + bx + c = 0

a = a, b = a^{2} + 1 and c = a

b^{2} – 4ac = (a^{2} + 1)^{2} - 4(a) (a)

= (a^{2})^{2} + 2a^{2} + 1 - 4a^{2}

= (a^{2})^{2} - 2a^{2} + 1

= (a^{2} - 1)^{2}

x = -b ± √(b² – 4ac)/2a

x = [ -a^{2} - 1 ± √(a^{2} - 1)^{2} ] / 2(a)

x = [ -a^{2} - 1 ± (a^{2} - 1) ] / 2a

x = (-a x = -2/2a x = -1/a |
x = (-a x = -2a x = -a |

Hence the solution is { -a, -1/a }.

**Example 2 :**

Solve using quadratic formula

36x^{2} – 12ax + (a^{2} - b^{2}) = 0

**Solution :**

a = 36, b = -12a and c = (a² - b²)

b^{2} – 4ac = (-12a)^{2} - 4(36)(a² - b²)

= 144a^{2} - 144a^{2} + 144b^{2}

= 144b^{2}

x = -b ± √(b² – 4ac)/2a

x = [ 12a ± √144b^{2} ] / 2(36)

x = [ 12a ± 12b ] / 72

x = (a ± b)/6

Hence the solution is (a + b)/6 and (a - b)/6.

**Example 3 :**

Solve by using quadratic formula

[(x–1)/(x+1)] + [(x–3)/(x–4)] = 10/3

**Solution :**

[(x - 4)(x – 1) + (x – 3)(x + 1)]/(x + 1)(x – 4) = 10/3

(x^{2} - 5x + 4 + x^{2} - 2x - 3)/(x^{2} - 3x - 4) = 10/3

(2x^{2} - 7x + 1)/(x^{2} - 3x - 4) = 10/3

3(2x^{2} - 7x + 1) = 10(x^{2} - 3x - 4)

10x^{2} - 6x^{2} - 30x + 21x - 40 - 3 = 0

4x^{2} - 9x - 43 = 0

a = 4, b = 9 and c = -43

b² – 4ac = (9)² – 4(4)(-43)

= (81 + 688)

= 769

x = -b ± √769)/2a

x = (-9 ± √769)/8

Hence the solution is { (-9 + √769)/8, (-9 - √769)/8 }.

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations using quadratic formula.

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