Solve left(m^2-1right)2^2+left(m-1right)2-4=0

$$\left( { m }^{ 2 } -1 \right) { 2 }^{ 2 } + \left( m-1 \right) 2-4 = 0$$

Solve for m

$$m = \frac{\sqrt{41} - 1}{4} \approx 1.350781059$$

$$m=\frac{-\sqrt{41}-1}{4}\approx -1.850781059$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 2 and get 4.

$$\left(m^{2}-1\right)\times 4+\left(m-1\right)\times 2-4=0$$

Use the distributive property to multiply m^{2}-1 by 4.

$$4m^{2}-4+\left(m-1\right)\times 2-4=0$$

Use the distributive property to multiply m-1 by 2.

$$4m^{2}-4+2m-2-4=0$$

Subtract 2 from -4 to get -6.

$$4m^{2}-6+2m-4=0$$

Subtract 4 from -6 to get -10.

$$4m^{2}-10+2m=0$$

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.

$$4m^{2}+2m-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-2±\sqrt{2^{2}-4\times 4\left(-10\right)}}{2\times 4}$$

Square 2.

$$m=\frac{-2±\sqrt{4-4\times 4\left(-10\right)}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-2±\sqrt{4-16\left(-10\right)}}{2\times 4}$$

Multiply -16 times -10.

$$m=\frac{-2±\sqrt{4+160}}{2\times 4}$$

Add 4 to 160.

$$m=\frac{-2±\sqrt{164}}{2\times 4}$$

Take the square root of 164.

$$m=\frac{-2±2\sqrt{41}}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{-2±2\sqrt{41}}{8}$$

Now solve the equation m=\frac{-2±2\sqrt{41}}{8} when ± is plus. Add -2 to 2\sqrt{41}.

$$m=\frac{2\sqrt{41}-2}{8}$$

Divide -2+2\sqrt{41} by 8.

$$m=\frac{\sqrt{41}-1}{4}$$

Now solve the equation m=\frac{-2±2\sqrt{41}}{8} when ± is minus. Subtract 2\sqrt{41} from -2.

$$m=\frac{-2\sqrt{41}-2}{8}$$

Divide -2-2\sqrt{41} by 8.

$$m=\frac{-\sqrt{41}-1}{4}$$

The equation is now solved.

$$m=\frac{\sqrt{41}-1}{4} m=\frac{-\sqrt{41}-1}{4}$$

Steps for Completing the Square

Calculate 2 to the power of 2 and get 4.

$$\left(m^{2}-1\right)\times 4+\left(m-1\right)\times 2-4=0$$

Use the distributive property to multiply m^{2}-1 by 4.

$$4m^{2}-4+\left(m-1\right)\times 2-4=0$$

Use the distributive property to multiply m-1 by 2.

$$4m^{2}-4+2m-2-4=0$$

Subtract 2 from -4 to get -6.

$$4m^{2}-6+2m-4=0$$

Subtract 4 from -6 to get -10.

$$4m^{2}-10+2m=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$4m^{2}+2m=10$$

Divide both sides by 4.

$$\frac{4m^{2}+2m}{4}=\frac{10}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}+\frac{2}{4}m=\frac{10}{4}$$

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

$$m^{2}+\frac{1}{2}m=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$m^{2}+\frac{1}{2}m=\frac{5}{2}$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{1}{2}m+\frac{1}{16}=\frac{5}{2}+\frac{1}{16}$$

Add \frac{5}{2} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}+\frac{1}{2}m+\frac{1}{16}=\frac{41}{16}$$

Factor m^{2}+\frac{1}{2}m+\frac{1}{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}.

$$\left(m+\frac{1}{4}\right)^{2}=\frac{41}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{4}\right)^{2}}=\sqrt{\frac{41}{16}}$$

Simplify.

$$m+\frac{1}{4}=\frac{\sqrt{41}}{4} m+\frac{1}{4}=-\frac{\sqrt{41}}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$m=\frac{\sqrt{41}-1}{4} m=\frac{-\sqrt{41}-1}{4}$$