Solve 0.1t^2=0.64+1.6(t-8)-4.9(t-8)^2

$$0.1 t ^ { 2 } = 0.64 + 1.6 ( t - 8 ) - 4.9 ( t - 8 ) ^ { 2 }$$

Solve for t

$$t=\frac{i\times 12\sqrt{5}}{25}+8\approx 8+1.073312629i$$

$$t=-\frac{i\times 12\sqrt{5}}{25}+8\approx 8-1.073312629i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1.6 by t-8.

$$0.1t^{2}=0.64+1.6t-12.8-4.9\left(t-8\right)^{2}$$

Subtract 12.8 from 0.64 to get -12.16.

$$0.1t^{2}=-12.16+1.6t-4.9\left(t-8\right)^{2}$$

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

$$0.1t^{2}=-12.16+1.6t-4.9\left(t^{2}-16t+64\right)$$

Use the distributive property to multiply -4.9 by t^{2}-16t+64.

$$0.1t^{2}=-12.16+1.6t-4.9t^{2}+78.4t-313.6$$

Combine 1.6t and 78.4t to get 80t.

$$0.1t^{2}=-12.16+80t-4.9t^{2}-313.6$$

Subtract 313.6 from -12.16 to get -325.76.

$$0.1t^{2}=-325.76+80t-4.9t^{2}$$

Subtract -325.76 from both sides.

$$0.1t^{2}-\left(-325.76\right)=80t-4.9t^{2}$$

The opposite of -325.76 is 325.76.

$$0.1t^{2}+325.76=80t-4.9t^{2}$$

Subtract 80t from both sides.

$$0.1t^{2}+325.76-80t=-4.9t^{2}$$

Add 4.9t^{2} to both sides.

$$0.1t^{2}+325.76-80t+4.9t^{2}=0$$

Combine 0.1t^{2} and 4.9t^{2} to get 5t^{2}.

$$5t^{2}+325.76-80t=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$5t^{2}-80t+325.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -80 for b, and 325.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 5\times 325.76}}{2\times 5}$$

Square -80.

$$t=\frac{-\left(-80\right)±\sqrt{6400-4\times 5\times 325.76}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-\left(-80\right)±\sqrt{6400-20\times 325.76}}{2\times 5}$$

Multiply -20 times 325.76.

$$t=\frac{-\left(-80\right)±\sqrt{6400-6515.2}}{2\times 5}$$

Add 6400 to -6515.2.

$$t=\frac{-\left(-80\right)±\sqrt{-115.2}}{2\times 5}$$

Take the square root of -115.2.

$$t=\frac{-\left(-80\right)±\frac{24\sqrt{5}i}{5}}{2\times 5}$$

The opposite of -80 is 80.

$$t=\frac{80±\frac{24\sqrt{5}i}{5}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{80±\frac{24\sqrt{5}i}{5}}{10}$$

Now solve the equation t=\frac{80±\frac{24\sqrt{5}i}{5}}{10} when ± is plus. Add 80 to \frac{24i\sqrt{5}}{5}.

$$t=\frac{\frac{24\sqrt{5}i}{5}+80}{10}$$

Divide 80+\frac{24i\sqrt{5}}{5} by 10.

$$t=\frac{12\sqrt{5}i}{25}+8$$

Now solve the equation t=\frac{80±\frac{24\sqrt{5}i}{5}}{10} when ± is minus. Subtract \frac{24i\sqrt{5}}{5} from 80.

$$t=\frac{-\frac{24\sqrt{5}i}{5}+80}{10}$$

Divide 80-\frac{24i\sqrt{5}}{5} by 10.

$$t=-\frac{12\sqrt{5}i}{25}+8$$

The equation is now solved.

$$t=\frac{12\sqrt{5}i}{25}+8 t=-\frac{12\sqrt{5}i}{25}+8$$

Steps for Completing the Square

Use the distributive property to multiply 1.6 by t-8.

$$0.1t^{2}=0.64+1.6t-12.8-4.9\left(t-8\right)^{2}$$

Subtract 12.8 from 0.64 to get -12.16.

$$0.1t^{2}=-12.16+1.6t-4.9\left(t-8\right)^{2}$$

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

$$0.1t^{2}=-12.16+1.6t-4.9\left(t^{2}-16t+64\right)$$

Use the distributive property to multiply -4.9 by t^{2}-16t+64.

$$0.1t^{2}=-12.16+1.6t-4.9t^{2}+78.4t-313.6$$

Combine 1.6t and 78.4t to get 80t.

$$0.1t^{2}=-12.16+80t-4.9t^{2}-313.6$$

Subtract 313.6 from -12.16 to get -325.76.

$$0.1t^{2}=-325.76+80t-4.9t^{2}$$

Subtract 80t from both sides.

$$0.1t^{2}-80t=-325.76-4.9t^{2}$$

Add 4.9t^{2} to both sides.

$$0.1t^{2}-80t+4.9t^{2}=-325.76$$

Combine 0.1t^{2} and 4.9t^{2} to get 5t^{2}.

$$5t^{2}-80t=-325.76$$

Divide both sides by 5.

$$\frac{5t^{2}-80t}{5}=-\frac{325.76}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\left(-\frac{80}{5}\right)t=-\frac{325.76}{5}$$

Divide -80 by 5.

$$t^{2}-16t=-\frac{325.76}{5}$$

Divide -325.76 by 5.

$$t^{2}-16t=-65.152$$

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.

$$t^{2}-16t+\left(-8\right)^{2}=-65.152+\left(-8\right)^{2}$$

Square -8.

$$t^{2}-16t+64=-65.152+64$$

Add -65.152 to 64.

$$t^{2}-16t+64=-1.152$$

Factor t^{2}-16t+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}.

$$\left(t-8\right)^{2}=-1.152$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-8\right)^{2}}=\sqrt{-1.152}$$

Simplify.

$$t-8=\frac{12\sqrt{5}i}{25} t-8=-\frac{12\sqrt{5}i}{25}$$

Add 8 to both sides of the equation.

$$t=\frac{12\sqrt{5}i}{25}+8 t=-\frac{12\sqrt{5}i}{25}+8$$