Solve left(-m^2-4m-3+m^2+12m-35right)^2=4

$${ \left(- { m }^{ 2 } -4m-3+ { m }^{ 2 } +12m-35 \right) }^{ 2 } = 4$$

Solve for m

$$m=5$$

$$m = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps Using the Quadratic Formula

Combine -4m and 12m to get 8m.

$$\left(-m^{2}+8m-3+m^{2}-35\right)^{2}=4$$

Subtract 35 from -3 to get -38.

$$\left(-m^{2}+8m-38+m^{2}\right)^{2}=4$$

Square -m^{2}+8m-38+m^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}-12m^{2}-76\left(-m^{2}\right)+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Multiply -76 and -1 to get 76.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}-12m^{2}+76m^{2}+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Combine -12m^{2} and 76m^{2} to get 64m^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Calculate -m^{2} to the power of 2 and get \left(m^{2}\right)^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+\left(m^{2}\right)^{2}+1444=4$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+m^{4}+1444=4$$

Combine m^{4} and m^{4} to get 2m^{4}.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+1444=4$$

Subtract 4 from both sides.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+1444-4=0$$

Subtract 4 from 1444 to get 1440.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+1440=0$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$2m^{4}+16m^{3}+2\left(-1\right)m^{4}+64m^{2}+16m\left(-1\right)m^{2}-608m+1440=0$$

Multiply 2 and -1 to get -2.

$$2m^{4}+16m^{3}-2m^{4}+64m^{2}+16m\left(-1\right)m^{2}-608m+1440=0$$

Combine 2m^{4} and -2m^{4} to get 0.

$$16m^{3}+64m^{2}+16m\left(-1\right)m^{2}-608m+1440=0$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$16m^{3}+64m^{2}+16m^{3}\left(-1\right)-608m+1440=0$$

Multiply 16 and -1 to get -16.

$$16m^{3}+64m^{2}-16m^{3}-608m+1440=0$$

Combine 16m^{3} and -16m^{3} to get 0.

$$64m^{2}-608m+1440=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, -608 for b, and 1440 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-608\right)±\sqrt{\left(-608\right)^{2}-4\times 64\times 1440}}{2\times 64}$$

Square -608.

$$m=\frac{-\left(-608\right)±\sqrt{369664-4\times 64\times 1440}}{2\times 64}$$

Multiply -4 times 64.

$$m=\frac{-\left(-608\right)±\sqrt{369664-256\times 1440}}{2\times 64}$$

Multiply -256 times 1440.

$$m=\frac{-\left(-608\right)±\sqrt{369664-368640}}{2\times 64}$$

Add 369664 to -368640.

$$m=\frac{-\left(-608\right)±\sqrt{1024}}{2\times 64}$$

Take the square root of 1024.

$$m=\frac{-\left(-608\right)±32}{2\times 64}$$

The opposite of -608 is 608.

$$m=\frac{608±32}{2\times 64}$$

Multiply 2 times 64.

$$m=\frac{608±32}{128}$$

Now solve the equation m=\frac{608±32}{128} when ± is plus. Add 608 to 32.

$$m=\frac{640}{128}$$

Divide 640 by 128.

$$m=5$$

Now solve the equation m=\frac{608±32}{128} when ± is minus. Subtract 32 from 608.

$$m=\frac{576}{128}$$

Reduce the fraction \frac{576}{128} to lowest terms by extracting and canceling out 64.

$$m=\frac{9}{2}$$

The equation is now solved.

$$m=5 m=\frac{9}{2}$$

Steps for Completing the Square

Combine -4m and 12m to get 8m.

$$\left(-m^{2}+8m-3+m^{2}-35\right)^{2}=4$$

Subtract 35 from -3 to get -38.

$$\left(-m^{2}+8m-38+m^{2}\right)^{2}=4$$

Square -m^{2}+8m-38+m^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}-12m^{2}-76\left(-m^{2}\right)+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Multiply -76 and -1 to get 76.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}-12m^{2}+76m^{2}+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Combine -12m^{2} and 76m^{2} to get 64m^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+\left(-m^{2}\right)^{2}+1444=4$$

Calculate -m^{2} to the power of 2 and get \left(m^{2}\right)^{2}.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+\left(m^{2}\right)^{2}+1444=4$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+m^{4}+1444=4$$

Combine m^{4} and m^{4} to get 2m^{4}.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m+1444=4$$

Subtract 1444 from both sides.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m=4-1444$$

Subtract 1444 from 4 to get -1440.

$$2m^{4}+16m^{3}+2\left(-m^{2}\right)m^{2}+64m^{2}+16m\left(-m^{2}\right)-608m=-1440$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$2m^{4}+16m^{3}+2\left(-1\right)m^{4}+64m^{2}+16m\left(-1\right)m^{2}-608m=-1440$$

Multiply 2 and -1 to get -2.

$$2m^{4}+16m^{3}-2m^{4}+64m^{2}+16m\left(-1\right)m^{2}-608m=-1440$$

Combine 2m^{4} and -2m^{4} to get 0.

$$16m^{3}+64m^{2}+16m\left(-1\right)m^{2}-608m=-1440$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$16m^{3}+64m^{2}+16m^{3}\left(-1\right)-608m=-1440$$

Multiply 16 and -1 to get -16.

$$16m^{3}+64m^{2}-16m^{3}-608m=-1440$$

Combine 16m^{3} and -16m^{3} to get 0.

$$64m^{2}-608m=-1440$$

Divide both sides by 64.

$$\frac{64m^{2}-608m}{64}=-\frac{1440}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$m^{2}+\left(-\frac{608}{64}\right)m=-\frac{1440}{64}$$

Reduce the fraction \frac{-608}{64} to lowest terms by extracting and canceling out 32.

$$m^{2}-\frac{19}{2}m=-\frac{1440}{64}$$

Reduce the fraction \frac{-1440}{64} to lowest terms by extracting and canceling out 32.

$$m^{2}-\frac{19}{2}m=-\frac{45}{2}$$

Divide -\frac{19}{2}, the coefficient of the x term, by 2 to get -\frac{19}{4}. Then add the square of -\frac{19}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{19}{2}m+\left(-\frac{19}{4}\right)^{2}=-\frac{45}{2}+\left(-\frac{19}{4}\right)^{2}$$

Square -\frac{19}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{19}{2}m+\frac{361}{16}=-\frac{45}{2}+\frac{361}{16}$$

Add -\frac{45}{2} to \frac{361}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{19}{2}m+\frac{361}{16}=\frac{1}{16}$$

Factor m^{2}-\frac{19}{2}m+\frac{361}{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}.

$$\left(m-\frac{19}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{19}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$m-\frac{19}{4}=\frac{1}{4} m-\frac{19}{4}=-\frac{1}{4}$$

Add \frac{19}{4} to both sides of the equation.

$$m=5 m=\frac{9}{2}$$