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

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

Solve for m

$$m=\sqrt{62}+8\approx 15.874007874$$

$$m=8-\sqrt{62}\approx 0.125992126$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2\right)^{2}.

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

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

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

Use the distributive property to multiply -3 by m^{2}+4.

$$4m^{2}-16m+16-3m^{2}-12=2$$

Combine 4m^{2} and -3m^{2} to get m^{2}.

$$m^{2}-16m+16-12=2$$

Subtract 12 from 16 to get 4.

$$m^{2}-16m+4=2$$

Subtract 2 from both sides.

$$m^{2}-16m+4-2=0$$

Subtract 2 from 4 to get 2.

$$m^{2}-16m+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -16.

$$m=\frac{-\left(-16\right)±\sqrt{256-4\times 2}}{2}$$

Multiply -4 times 2.

$$m=\frac{-\left(-16\right)±\sqrt{256-8}}{2}$$

Add 256 to -8.

$$m=\frac{-\left(-16\right)±\sqrt{248}}{2}$$

Take the square root of 248.

$$m=\frac{-\left(-16\right)±2\sqrt{62}}{2}$$

The opposite of -16 is 16.

$$m=\frac{16±2\sqrt{62}}{2}$$

Now solve the equation m=\frac{16±2\sqrt{62}}{2} when ± is plus. Add 16 to 2\sqrt{62}.

$$m=\frac{2\sqrt{62}+16}{2}$$

Divide 16+2\sqrt{62} by 2.

$$m=\sqrt{62}+8$$

Now solve the equation m=\frac{16±2\sqrt{62}}{2} when ± is minus. Subtract 2\sqrt{62} from 16.

$$m=\frac{16-2\sqrt{62}}{2}$$

Divide 16-2\sqrt{62} by 2.

$$m=8-\sqrt{62}$$

The equation is now solved.

$$m=\sqrt{62}+8 m=8-\sqrt{62}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2\right)^{2}.

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

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

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

Use the distributive property to multiply -3 by m^{2}+4.

$$4m^{2}-16m+16-3m^{2}-12=2$$

Combine 4m^{2} and -3m^{2} to get m^{2}.

$$m^{2}-16m+16-12=2$$

Subtract 12 from 16 to get 4.

$$m^{2}-16m+4=2$$

Subtract 4 from both sides.

$$m^{2}-16m=2-4$$

Subtract 4 from 2 to get -2.

$$m^{2}-16m=-2$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-16m+\left(-8\right)^{2}=-2+\left(-8\right)^{2}$$

Square -8.

$$m^{2}-16m+64=-2+64$$

Add -2 to 64.

$$m^{2}-16m+64=62$$

Factor m^{2}-16m+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-8\right)^{2}=62$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-8\right)^{2}}=\sqrt{62}$$

Simplify.

$$m-8=\sqrt{62} m-8=-\sqrt{62}$$

Add 8 to both sides of the equation.

$$m=\sqrt{62}+8 m=8-\sqrt{62}$$