Solve left(x^2-x+1right)6=4x^2-x+5

$$\left( { x }^{ 2 } -x+1 \right) 6=4 { x }^{ 2 } -x+5$$

Solve for x

$$x = \frac{\sqrt{17} + 5}{4} \approx 2.280776406$$

$$x=\frac{5-\sqrt{17}}{4}\approx 0.219223594$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x^{2}-x+1 by 6.

$$6x^{2}-6x+6=4x^{2}-x+5$$

Subtract 4x^{2} from both sides.

$$6x^{2}-6x+6-4x^{2}=-x+5$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}-6x+6=-x+5$$

Add x to both sides.

$$2x^{2}-6x+6+x=5$$

Combine -6x and x to get -5x.

$$2x^{2}-5x+6=5$$

Subtract 5 from both sides.

$$2x^{2}-5x+6-5=0$$

Subtract 5 from 6 to get 1.

$$2x^{2}-5x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2}}{2\times 2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8}}{2\times 2}$$

Add 25 to -8.

$$x=\frac{-\left(-5\right)±\sqrt{17}}{2\times 2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{17}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5±\sqrt{17}}{4}$$

Now solve the equation x=\frac{5±\sqrt{17}}{4} when ± is plus. Add 5 to \sqrt{17}.

$$x=\frac{\sqrt{17}+5}{4}$$

Now solve the equation x=\frac{5±\sqrt{17}}{4} when ± is minus. Subtract \sqrt{17} from 5.

$$x=\frac{5-\sqrt{17}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+5}{4} x=\frac{5-\sqrt{17}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply x^{2}-x+1 by 6.

$$6x^{2}-6x+6=4x^{2}-x+5$$

Subtract 4x^{2} from both sides.

$$6x^{2}-6x+6-4x^{2}=-x+5$$

Combine 6x^{2} and -4x^{2} to get 2x^{2}.

$$2x^{2}-6x+6=-x+5$$

Add x to both sides.

$$2x^{2}-6x+6+x=5$$

Combine -6x and x to get -5x.

$$2x^{2}-5x+6=5$$

Subtract 6 from both sides.

$$2x^{2}-5x=5-6$$

Subtract 6 from 5 to get -1.

$$2x^{2}-5x=-1$$

Divide both sides by 2.

$$\frac{2x^{2}-5x}{2}=-\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{5}{2}x=-\frac{1}{2}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{1}{2}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{1}{2}+\frac{25}{16}$$

Add -\frac{1}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{17}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{17}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{17}}{4} x-\frac{5}{4}=-\frac{\sqrt{17}}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{\sqrt{17}+5}{4} x=\frac{5-\sqrt{17}}{4}$$