Solve (m-1)^2+3(m-5)^2=16

$$( m - 1 ) ^ { 2 } + 3 ( m - 5 ) ^ { 2 } = 16$$

Solve for m

$$m=3$$

$$m=5$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-1\right)^{2}.

$$m^{2}-2m+1+3\left(m-5\right)^{2}=16$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-5\right)^{2}.

$$m^{2}-2m+1+3\left(m^{2}-10m+25\right)=16$$

Use the distributive property to multiply 3 by m^{2}-10m+25.

$$m^{2}-2m+1+3m^{2}-30m+75=16$$

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

$$4m^{2}-2m+1-30m+75=16$$

Combine -2m and -30m to get -32m.

$$4m^{2}-32m+1+75=16$$

Add 1 and 75 to get 76.

$$4m^{2}-32m+76=16$$

Subtract 16 from both sides.

$$4m^{2}-32m+76-16=0$$

Subtract 16 from 76 to get 60.

$$4m^{2}-32m+60=0$$

Divide both sides by 4.

$$m^{2}-8m+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as m^{2}+am+bm+15. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\times 15=15$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -8.

$$a=-5 b=-3$$

Rewrite m^{2}-8m+15 as \left(m^{2}-5m\right)+\left(-3m+15\right).

$$\left(m^{2}-5m\right)+\left(-3m+15\right)$$

Factor out m in the first and -3 in the second group.

$$m\left(m-5\right)-3\left(m-5\right)$$

Factor out common term m-5 by using distributive property.

$$\left(m-5\right)\left(m-3\right)$$

To find equation solutions, solve m-5=0 and m-3=0.

$$m=5 m=3$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-1\right)^{2}.

$$m^{2}-2m+1+3\left(m-5\right)^{2}=16$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-5\right)^{2}.

$$m^{2}-2m+1+3\left(m^{2}-10m+25\right)=16$$

Use the distributive property to multiply 3 by m^{2}-10m+25.

$$m^{2}-2m+1+3m^{2}-30m+75=16$$

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

$$4m^{2}-2m+1-30m+75=16$$

Combine -2m and -30m to get -32m.

$$4m^{2}-32m+1+75=16$$

Add 1 and 75 to get 76.

$$4m^{2}-32m+76=16$$

Subtract 16 from both sides.

$$4m^{2}-32m+76-16=0$$

Subtract 16 from 76 to get 60.

$$4m^{2}-32m+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -32 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 4\times 60}}{2\times 4}$$

Square -32.

$$m=\frac{-\left(-32\right)±\sqrt{1024-4\times 4\times 60}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-\left(-32\right)±\sqrt{1024-16\times 60}}{2\times 4}$$

Multiply -16 times 60.

$$m=\frac{-\left(-32\right)±\sqrt{1024-960}}{2\times 4}$$

Add 1024 to -960.

$$m=\frac{-\left(-32\right)±\sqrt{64}}{2\times 4}$$

Take the square root of 64.

$$m=\frac{-\left(-32\right)±8}{2\times 4}$$

The opposite of -32 is 32.

$$m=\frac{32±8}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{32±8}{8}$$

Now solve the equation m=\frac{32±8}{8} when ± is plus. Add 32 to 8.

$$m=\frac{40}{8}$$

Divide 40 by 8.

$$m=5$$

Now solve the equation m=\frac{32±8}{8} when ± is minus. Subtract 8 from 32.

$$m=\frac{24}{8}$$

Divide 24 by 8.

$$m=3$$

The equation is now solved.

$$m=5 m=3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-1\right)^{2}.

$$m^{2}-2m+1+3\left(m-5\right)^{2}=16$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-5\right)^{2}.

$$m^{2}-2m+1+3\left(m^{2}-10m+25\right)=16$$

Use the distributive property to multiply 3 by m^{2}-10m+25.

$$m^{2}-2m+1+3m^{2}-30m+75=16$$

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

$$4m^{2}-2m+1-30m+75=16$$

Combine -2m and -30m to get -32m.

$$4m^{2}-32m+1+75=16$$

Add 1 and 75 to get 76.

$$4m^{2}-32m+76=16$$

Subtract 76 from both sides.

$$4m^{2}-32m=16-76$$

Subtract 76 from 16 to get -60.

$$4m^{2}-32m=-60$$

Divide both sides by 4.

$$\frac{4m^{2}-32m}{4}=-\frac{60}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}+\left(-\frac{32}{4}\right)m=-\frac{60}{4}$$

Divide -32 by 4.

$$m^{2}-8m=-\frac{60}{4}$$

Divide -60 by 4.

$$m^{2}-8m=-15$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-8m+\left(-4\right)^{2}=-15+\left(-4\right)^{2}$$

Square -4.

$$m^{2}-8m+16=-15+16$$

Add -15 to 16.

$$m^{2}-8m+16=1$$

Factor m^{2}-8m+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m-4=1 m-4=-1$$

Add 4 to both sides of the equation.

$$m=5 m=3$$