Solve left(t-6right)^2+t^2+80=left(t-2right)^2+left(2t-8right)^2

$${ \left(t-6 \right) }^{ 2 } + { t }^{ 2 } +80 = { \left(t-2 \right) }^{ 2 } + { \left(2t-8 \right) }^{ 2 }$$

Solve for t

$$t=4\sqrt{2}+4\approx 9.656854249$$

$$t=4-4\sqrt{2}\approx -1.656854249$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-6\right)^{2}.

$$t^{2}-12t+36+t^{2}+80=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

Combine t^{2} and t^{2} to get 2t^{2}.

$$2t^{2}-12t+36+80=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

Add 36 and 80 to get 116.

$$2t^{2}-12t+116=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

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

$$2t^{2}-12t+116=t^{2}-4t+4+\left(2t-8\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2t-8\right)^{2}.

$$2t^{2}-12t+116=t^{2}-4t+4+4t^{2}-32t+64$$

Combine t^{2} and 4t^{2} to get 5t^{2}.

$$2t^{2}-12t+116=5t^{2}-4t+4-32t+64$$

Combine -4t and -32t to get -36t.

$$2t^{2}-12t+116=5t^{2}-36t+4+64$$

Add 4 and 64 to get 68.

$$2t^{2}-12t+116=5t^{2}-36t+68$$

Subtract 5t^{2} from both sides.

$$2t^{2}-12t+116-5t^{2}=-36t+68$$

Combine 2t^{2} and -5t^{2} to get -3t^{2}.

$$-3t^{2}-12t+116=-36t+68$$

Add 36t to both sides.

$$-3t^{2}-12t+116+36t=68$$

Combine -12t and 36t to get 24t.

$$-3t^{2}+24t+116=68$$

Subtract 68 from both sides.

$$-3t^{2}+24t+116-68=0$$

Subtract 68 from 116 to get 48.

$$-3t^{2}+24t+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 24 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-24±\sqrt{24^{2}-4\left(-3\right)\times 48}}{2\left(-3\right)}$$

Square 24.

$$t=\frac{-24±\sqrt{576-4\left(-3\right)\times 48}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$t=\frac{-24±\sqrt{576+12\times 48}}{2\left(-3\right)}$$

Multiply 12 times 48.

$$t=\frac{-24±\sqrt{576+576}}{2\left(-3\right)}$$

Add 576 to 576.

$$t=\frac{-24±\sqrt{1152}}{2\left(-3\right)}$$

Take the square root of 1152.

$$t=\frac{-24±24\sqrt{2}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$t=\frac{-24±24\sqrt{2}}{-6}$$

Now solve the equation t=\frac{-24±24\sqrt{2}}{-6} when ± is plus. Add -24 to 24\sqrt{2}.

$$t=\frac{24\sqrt{2}-24}{-6}$$

Divide -24+24\sqrt{2} by -6.

$$t=4-4\sqrt{2}$$

Now solve the equation t=\frac{-24±24\sqrt{2}}{-6} when ± is minus. Subtract 24\sqrt{2} from -24.

$$t=\frac{-24\sqrt{2}-24}{-6}$$

Divide -24-24\sqrt{2} by -6.

$$t=4\sqrt{2}+4$$

The equation is now solved.

$$t=4-4\sqrt{2} t=4\sqrt{2}+4$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-6\right)^{2}.

$$t^{2}-12t+36+t^{2}+80=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

Combine t^{2} and t^{2} to get 2t^{2}.

$$2t^{2}-12t+36+80=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

Add 36 and 80 to get 116.

$$2t^{2}-12t+116=\left(t-2\right)^{2}+\left(2t-8\right)^{2}$$

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

$$2t^{2}-12t+116=t^{2}-4t+4+\left(2t-8\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2t-8\right)^{2}.

$$2t^{2}-12t+116=t^{2}-4t+4+4t^{2}-32t+64$$

Combine t^{2} and 4t^{2} to get 5t^{2}.

$$2t^{2}-12t+116=5t^{2}-4t+4-32t+64$$

Combine -4t and -32t to get -36t.

$$2t^{2}-12t+116=5t^{2}-36t+4+64$$

Add 4 and 64 to get 68.

$$2t^{2}-12t+116=5t^{2}-36t+68$$

Subtract 5t^{2} from both sides.

$$2t^{2}-12t+116-5t^{2}=-36t+68$$

Combine 2t^{2} and -5t^{2} to get -3t^{2}.

$$-3t^{2}-12t+116=-36t+68$$

Add 36t to both sides.

$$-3t^{2}-12t+116+36t=68$$

Combine -12t and 36t to get 24t.

$$-3t^{2}+24t+116=68$$

Subtract 116 from both sides.

$$-3t^{2}+24t=68-116$$

Subtract 116 from 68 to get -48.

$$-3t^{2}+24t=-48$$

Divide both sides by -3.

$$\frac{-3t^{2}+24t}{-3}=-\frac{48}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$t^{2}+\frac{24}{-3}t=-\frac{48}{-3}$$

Divide 24 by -3.

$$t^{2}-8t=-\frac{48}{-3}$$

Divide -48 by -3.

$$t^{2}-8t=16$$

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.

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

Square -4.

$$t^{2}-8t+16=16+16$$

Add 16 to 16.

$$t^{2}-8t+16=32$$

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

$$\left(t-4\right)^{2}=32$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-4\right)^{2}}=\sqrt{32}$$

Simplify.

$$t-4=4\sqrt{2} t-4=-4\sqrt{2}$$

Add 4 to both sides of the equation.

$$t=4\sqrt{2}+4 t=4-4\sqrt{2}$$