How to solve the quadratic equation

Quadratic equation Math Problem

Solve quadratic equation by factorization √(5x² – 6x + 8) – √(5x² – 6x – 7) = 1

Solving the quadratic equation by factorization

Let, u = √(5x² – 6x + 8)

and v = √(5x² – 6x – 7)

then

u – v = 1……….eq(1)

u² – v² = 5x² – 6x + 8 – (5x² – 6x – 7)

⇒ u² – v² = 5x² – 6x + 8 – 5x² + 6x + 7

⇒ u² – v² = 15

we know, u² – v² = (u + v)(u – v)

⇒ 15 = (u + v) × 1

⇒ u + v = 15……….eq(2)

Add equation 1 and equation 2

u – v + u + v = 1 + 15

⇒ 2u = 16

⇒ u = 8

then, v = 7

When u = 8

u² = 64 = 5x² – 6x + 8

⇒ 5x² – 6x – 56 = 0

⇒ 5x² – 20x + 14x – 56 = 0

We can solve the quadratic equation by factorization

so, 5(x – 4) + 14(x – 4) = 0

⇒ (x – 4)(5x + 14) = 0

⇒ x = 4 or x = – 14/5

When u = 7

v² = 49 = 5x² – 6x – 7

⇒ 5x² – 6x – 56 = 0

⇒ x = 4 or x = – 14/5

Now the so the solutions are x = 4 or x = – 14/5