Question

Solve using the quadratic formula. 2x2=8x-7

Answer 1

Answer:

$\boxed{x=\frac{4\pm\sqrt{2}}{2}}$

Step-by-step 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 $\boxed{x=\frac{-b\pm\sqrt{b^2-4ac} }{2a} }$.

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.

$x=\frac{8\pm\sqrt{(-8)^2-4(2)(7)}}{2(2)} \\\\x=\frac{8\pm\sqrt{64-4(14)}}{4}\\\\x=\frac{8\pm\sqrt{64-56}}{4} \\\\x=\frac{8\pm\sqrt{8}}{4}\\\\x= 2\pm\frac{\sqrt{8}}{4}\\ \\x=2\pm\frac{\sqrt{2}}{2}$

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

$x=2+\frac{\sqrt{2} }{2}\\\\x=\frac{4+\sqrt{2}}{2}$

Operation 2: Subtraction

$x=2-\frac{\sqrt{2}}{2}\\ \\x=\frac{4-\sqrt{2}}{2}$

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

$\boxed{x=\frac{4\pm\sqrt{2}}{2}}$