Solve 2.5=2x^2-8x+6

$$2.5 = 2 x ^ { 2 } - 8 x + 6$$

Solve for x

$$x=0.5$$

$$x=3.5$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-8x+6=2.5$$

Subtract 2.5 from both sides.

$$2x^{2}-8x+6-2.5=0$$

Subtract 2.5 from 6 to get 3.5.

$$2x^{2}-8x+3.5=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}-8x+\frac{7}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -8 for b, and \frac{7}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\times \frac{7}{2}}}{2\times 2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\times \frac{7}{2}}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\times \frac{7}{2}}}{2\times 2}$$

Multiply -8 times \frac{7}{2}.

$$x=\frac{-\left(-8\right)±\sqrt{64-28}}{2\times 2}$$

Add 64 to -28.

$$x=\frac{-\left(-8\right)±\sqrt{36}}{2\times 2}$$

Take the square root of 36.

$$x=\frac{-\left(-8\right)±6}{2\times 2}$$

The opposite of -8 is 8.

$$x=\frac{8±6}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{8±6}{4}$$

Now solve the equation x=\frac{8±6}{4} when ± is plus. Add 8 to 6.

$$x=\frac{14}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{2}$$

Now solve the equation x=\frac{8±6}{4} when ± is minus. Subtract 6 from 8.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=\frac{7}{2} x=\frac{1}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}-8x+6=2.5$$

Subtract 6 from both sides.

$$2x^{2}-8x=2.5-6$$

Subtract 6 from 2.5 to get -3.5.

$$2x^{2}-8x=-3.5$$

Divide both sides by 2.

$$\frac{2x^{2}-8x}{2}=-\frac{3.5}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{8}{2}\right)x=-\frac{3.5}{2}$$

Divide -8 by 2.

$$x^{2}-4x=-\frac{3.5}{2}$$

Divide -3.5 by 2.

$$x^{2}-4x=-1.75$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=-1.75+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-1.75+4$$

Add -1.75 to 4.

$$x^{2}-4x+4=2.25$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=2.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{2.25}$$

Simplify.

$$x-2=\frac{3}{2} x-2=-\frac{3}{2}$$

Add 2 to both sides of the equation.

$$x=\frac{7}{2} x=\frac{1}{2}$$