SOLVING QUADRATIC EQUATIONS BY QUADRATIC FORMULA 

The standard form of a quadratic equation is

ax2 + bx + c = 0

Quadratic Formula :

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.

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⁄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

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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