Question

Use the quadratic formula to solve the equation.

4x^2 - 10x + 5 = 0
Enter your answers, in simplified radical form.

X=_____ or X=_____​

Answer 1

`\large\boxed{x=(5-\sqrt5)/(4),\ x=(5+\sqrt5)/(4)}`

Explanation:

`\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=(-b)/(2a)\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=(-b\pm√(b^2-4ac))/(2a)\\\\==========================================`

`\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=(-(-10)\pm√(20))/(2(4))=(10\pm√(4\cdot5))/(8)=(10\pm\sqrt4\cdot\sqrt5)/(8)=(10\pm2\sqrt5)/(8)\\\\=(2(5\pm\sqrt5))/(8)=(5\pm\sqrt5)/(4)`

Answer 2

`x = ( 5 - √( 5) )/(4) \: or \: \: x = ( 5 + √( 5) )/(4)`

EXPLANATION

The given quadratic equation is

`4 {x}^(2) - 10x + 5 = 0`

We compare this to

`a {x}^(2) + bx + c = 0`

to get a=4, b=-10, and c=5.

The quadratic formula is given by

`x = \frac{ - b \pm \sqrt{ {b}^(2) - 4ac} }{2a}`

We substitute these values into the formula to get:

`x = \frac{ - - 10 \pm \sqrt{ {( - 10)}^(2) - 4(4)(5)} }{2(4)}`

This implies that

`x = ( 10 \pm √( 100 - 80) )/(8)`

`x = ( 10 \pm √( 20) )/(8)`

`x = ( 10 \pm2 √( 5) )/(8)`

`x = ( 5 \pm √( 5) )/(4)`

The solutions are:

`x = ( 5 - √( 5) )/(4) \: or \: \: x = ( 5 + √( 5) )/(4)`