Solve (m-4)^2+4(m-3)=5

$$( m - 4 ) ^ { 2 } + 4 ( m - 3 ) = 5$$

Solve for m

$$m=\sqrt{5}+2\approx 4.236067977$$

$$m=2-\sqrt{5}\approx -0.236067977$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-4\right)^{2}.

$$m^{2}-8m+16+4\left(m-3\right)=5$$

Use the distributive property to multiply 4 by m-3.

$$m^{2}-8m+16+4m-12=5$$

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

$$m^{2}-4m+16-12=5$$

Subtract 12 from 16 to get 4.

$$m^{2}-4m+4=5$$

Subtract 5 from both sides.

$$m^{2}-4m+4-5=0$$

Subtract 5 from 4 to get -1.

$$m^{2}-4m-1=0$$

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

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

Square -4.

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

Multiply -4 times -1.

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

Add 16 to 4.

$$m=\frac{-\left(-4\right)±\sqrt{20}}{2}$$

Take the square root of 20.

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

The opposite of -4 is 4.

$$m=\frac{4±2\sqrt{5}}{2}$$

Now solve the equation m=\frac{4±2\sqrt{5}}{2} when ± is plus. Add 4 to 2\sqrt{5}.

$$m=\frac{2\sqrt{5}+4}{2}$$

Divide 4+2\sqrt{5} by 2.

$$m=\sqrt{5}+2$$

Now solve the equation m=\frac{4±2\sqrt{5}}{2} when ± is minus. Subtract 2\sqrt{5} from 4.

$$m=\frac{4-2\sqrt{5}}{2}$$

Divide 4-2\sqrt{5} by 2.

$$m=2-\sqrt{5}$$

The equation is now solved.

$$m=\sqrt{5}+2 m=2-\sqrt{5}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-4\right)^{2}.

$$m^{2}-8m+16+4\left(m-3\right)=5$$

Use the distributive property to multiply 4 by m-3.

$$m^{2}-8m+16+4m-12=5$$

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

$$m^{2}-4m+16-12=5$$

Subtract 12 from 16 to get 4.

$$m^{2}-4m+4=5$$

Factor m^{2}-4m+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}.

$$\left(m-2\right)^{2}=5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-2\right)^{2}}=\sqrt{5}$$

Simplify.

$$m-2=\sqrt{5} m-2=-\sqrt{5}$$

Add 2 to both sides of the equation.

$$m=\sqrt{5}+2 m=2-\sqrt{5}$$