Solve (t+40)^2=50(5t^2-28t+40)

$$( t + 40 ) ^ { 2 } = 50 ( 5 t ^ { 2 } - 28 t + 40 )$$

Solve for t

$$t = \frac{80 \sqrt{70} + 740}{249} \approx 5.659951892$$

$$t=\frac{740-80\sqrt{70}}{249}\approx 0.283823208$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(t+40\right)^{2}.

$$t^{2}+80t+1600=50\left(5t^{2}-28t+40\right)$$

Use the distributive property to multiply 50 by 5t^{2}-28t+40.

$$t^{2}+80t+1600=250t^{2}-1400t+2000$$

Subtract 250t^{2} from both sides.

$$t^{2}+80t+1600-250t^{2}=-1400t+2000$$

Combine t^{2} and -250t^{2} to get -249t^{2}.

$$-249t^{2}+80t+1600=-1400t+2000$$

Add 1400t to both sides.

$$-249t^{2}+80t+1600+1400t=2000$$

Combine 80t and 1400t to get 1480t.

$$-249t^{2}+1480t+1600=2000$$

Subtract 2000 from both sides.

$$-249t^{2}+1480t+1600-2000=0$$

Subtract 2000 from 1600 to get -400.

$$-249t^{2}+1480t-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -249 for a, 1480 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-1480±\sqrt{1480^{2}-4\left(-249\right)\left(-400\right)}}{2\left(-249\right)}$$

Square 1480.

$$t=\frac{-1480±\sqrt{2190400-4\left(-249\right)\left(-400\right)}}{2\left(-249\right)}$$

Multiply -4 times -249.

$$t=\frac{-1480±\sqrt{2190400+996\left(-400\right)}}{2\left(-249\right)}$$

Multiply 996 times -400.

$$t=\frac{-1480±\sqrt{2190400-398400}}{2\left(-249\right)}$$

Add 2190400 to -398400.

$$t=\frac{-1480±\sqrt{1792000}}{2\left(-249\right)}$$

Take the square root of 1792000.

$$t=\frac{-1480±160\sqrt{70}}{2\left(-249\right)}$$

Multiply 2 times -249.

$$t=\frac{-1480±160\sqrt{70}}{-498}$$

Now solve the equation t=\frac{-1480±160\sqrt{70}}{-498} when ± is plus. Add -1480 to 160\sqrt{70}.

$$t=\frac{160\sqrt{70}-1480}{-498}$$

Divide -1480+160\sqrt{70} by -498.

$$t=\frac{740-80\sqrt{70}}{249}$$

Now solve the equation t=\frac{-1480±160\sqrt{70}}{-498} when ± is minus. Subtract 160\sqrt{70} from -1480.

$$t=\frac{-160\sqrt{70}-1480}{-498}$$

Divide -1480-160\sqrt{70} by -498.

$$t=\frac{80\sqrt{70}+740}{249}$$

The equation is now solved.

$$t=\frac{740-80\sqrt{70}}{249} t=\frac{80\sqrt{70}+740}{249}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(t+40\right)^{2}.

$$t^{2}+80t+1600=50\left(5t^{2}-28t+40\right)$$

Use the distributive property to multiply 50 by 5t^{2}-28t+40.

$$t^{2}+80t+1600=250t^{2}-1400t+2000$$

Subtract 250t^{2} from both sides.

$$t^{2}+80t+1600-250t^{2}=-1400t+2000$$

Combine t^{2} and -250t^{2} to get -249t^{2}.

$$-249t^{2}+80t+1600=-1400t+2000$$

Add 1400t to both sides.

$$-249t^{2}+80t+1600+1400t=2000$$

Combine 80t and 1400t to get 1480t.

$$-249t^{2}+1480t+1600=2000$$

Subtract 1600 from both sides.

$$-249t^{2}+1480t=2000-1600$$

Subtract 1600 from 2000 to get 400.

$$-249t^{2}+1480t=400$$

Divide both sides by -249.

$$\frac{-249t^{2}+1480t}{-249}=\frac{400}{-249}$$

Dividing by -249 undoes the multiplication by -249.

$$t^{2}+\frac{1480}{-249}t=\frac{400}{-249}$$

Divide 1480 by -249.

$$t^{2}-\frac{1480}{249}t=\frac{400}{-249}$$

Divide 400 by -249.

$$t^{2}-\frac{1480}{249}t=-\frac{400}{249}$$

Divide -\frac{1480}{249}, the coefficient of the x term, by 2 to get -\frac{740}{249}. Then add the square of -\frac{740}{249} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1480}{249}t+\left(-\frac{740}{249}\right)^{2}=-\frac{400}{249}+\left(-\frac{740}{249}\right)^{2}$$

Square -\frac{740}{249} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1480}{249}t+\frac{547600}{62001}=-\frac{400}{249}+\frac{547600}{62001}$$

Add -\frac{400}{249} to \frac{547600}{62001} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1480}{249}t+\frac{547600}{62001}=\frac{448000}{62001}$$

Factor t^{2}-\frac{1480}{249}t+\frac{547600}{62001}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{740}{249}\right)^{2}=\frac{448000}{62001}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{740}{249}\right)^{2}}=\sqrt{\frac{448000}{62001}}$$

Simplify.

$$t-\frac{740}{249}=\frac{80\sqrt{70}}{249} t-\frac{740}{249}=-\frac{80\sqrt{70}}{249}$$

Add \frac{740}{249} to both sides of the equation.

$$t=\frac{80\sqrt{70}+740}{249} t=\frac{740-80\sqrt{70}}{249}$$