The standard form of a quadratic equation is

ax2 + bx + c = 0

The above formula can be used to solve a quadratic equation in standard form. If the given quadratic equation is not in standard form, convert it to standard form and use the above formula and solve.

Example 1 :

x2 – 5x – 24 = 0

Solution :

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

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

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

x = 8  or  -3

Example 2 :

x2 – 7x + 12 = 0

Solution :

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 :

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

Substitute the above values into the quadratic formula.

Example 4 :

15x2 – 11x + 2  =  0

Solution :

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⁄ₓ⁵⁄₂

Multiply both sides by 2x.

2x[x + ¹⁄ₓ] = 2x[⁵⁄₂]

2x2 + 2x(¹⁄ₓ) = 5x

2x2 + 2 = 5x

2x2 - 5x + 2 = 0

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

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

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

a = a, b = -(a2 + 1), c = a

Substitute the above values into the quadratic formula.

Example 9 :

3a2x2 - abx - 2b2 = 0

Solution :

a = 3a2, b = -ab, c = -2b2

Substitute the above values into the quadratic formula.

Example 10 :

36x2 – 12ax + (a2 - b2) = 0

Solution :

a = 36, b = -12a, c = a2 - b2

Substitute the above values into the quadratic formula.

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