SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA 

Solve each of the following quadratic equations using the quadratic formula.

Example 1 :

x² – 5x – 24 = 0

Solution :

Comparing the given equation with ax² + bx + c = 0, we get

a = 1, b = -5, c = -24

Quadratic Formula :

Substitute a = 1, b = -5 and c = -24.

x = 8  or  -3

Example 2 :

x² – 7x + 12 = 0

Solution :

From the given quadratic equation,

a = 1, b = -7, c = 12

Substitute the above values into the quadratic formula.

x = 4  or  3

Example 3 :

x² – 2x - 5  =  0

Solution :

From the given quadratic equation,

a = 1, b = -2, c = -5

Substitute the above values into the quadratic formula.

Example 4 :

15x² – 11x + 2  =  0

Solution :

From the given quadratic equation,

a = 15, b = -11, c = 2

Substitute the above values into the quadratic formula.

Example 5 :

x + ¹⁄ₓ = 2½

Solution :

x + ¹⁄ₓ = 2½

x + ¹⁄ₓ = ⁵⁄₂

Multiply both sides by 2x.

2x[x + ¹⁄ₓ] = 2x[⁵⁄₂]

2x² + 2x(¹⁄ₓ) = 5x

2x² + 2 = 5x

2x² - 5x + 2 = 0

From the given quadratic equation,

a = 2, b = -5, c = 2

Substitute the above values into the quadratic formula.

Example 6 :

(x + 3)² - 81 = 0

Solution :

(x + 3)² - 81 = 0

(x + 3)(x + 3) - 81 = 0

x² + 3x + 3x + 9 - 81 = 0

x² + 6x - 72 = 0

From the given quadratic equation,

a = 1, b = 6, c = -72

Substitute the above values into the quadratic formula.

x = 6  or  -12

Example 7 :

Solution :

4x² - 9x - 43 = 0

From the given quadratic equation,

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

Substitute the above values into the quadratic formula.

Example 8 :

 a(x² + 1) = x(a² + 1)

Solution :

 a(x² + 1) = x(a² + 1)

 ax² + a = xa² + x

 ax² + a - xa² - x = 0

 ax² - xa² - x + a = 0

 ax² - (a² + 1)x + a = 0

From the given quadratic equation,

a = a, b = -(a² + 1), c = a

Substitute the above values into the quadratic formula.

Example 9 :

3a²x² - abx - 2b² = 0   

Solution :

From the given quadratic equation,

a = 3a², b = -ab, c = -2b²

Substitute the above values into the quadratic formula.

Example 10 :

36x² – 12ax + (a² - b²) = 0

Solution :

From the given quadratic equation,

a = 36, b = -12a, c = a² - b²

Substitute the above values into the quadratic formula.

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