Question

Solve using the quadratic formula.

2x2=8x-7

Answer 1

`\boxed{x=(4\pm√(2))/(2)}`

Explanation:

Part 1: Rewriting equation to match ax² + bx + c = 0 (quadratic function)

The given equation is not written in quadratic form. To rewrite the equation:

• All values need to be on the left side of the equation and set equal to zero.

To overcome this difficulty, follow these mathematical steps:

`2x^2=8x-7\\2x^2-8x=-7\\2x^2-8x+7=0`

Subtract 8x from both sides of the equation to rearrange it to the left side. Then, add 7 to rearrange it as well. Finally, set the three values on the left of the equation equal to zero.

Part 2: Using the quadratic formula

The quadratic formula is defined as
`\boxed{x=(-b\pm√(b^2-4ac) )/(2a) }`.

Using the parent quadratic function, the values are easy to find in the given equation.
`\boxed{a=2, b=-8, c=7}`

Substitute these values into the quadratic formula and solve for x.

`x=(8\pm√((-8)^2-4(2)(7)))/(2(2)) \\\\x=(8\pm√(64-4(14)))/(4)\\\\x=(8\pm√(64-56))/(4) \\\\x=(8\pm√(8))/(4)\\\\x= 2\pm(√(8))/(4)\\ \\x=2\pm(√(2))/(2)`

Part 3: Solving for x with the values from the quadratic formula

Now that x is set equal to the simplified version of the equation, the operations have to be followed through with.

This equation will have two zeros/roots to solve for by setting x equal to zero.

Operation 1: Addition

`x=2+(√(2) )/(2)\\\\x=(4+√(2))/(2)`

Operation 2: Subtraction

`x=2-(√(2))/(2)\\ \\x=(4-√(2))/(2)`

Because both values are the exact same (minus the operations), the roots can be simplified even further to one value:

`\boxed{x=(4\pm√(2))/(2)}`