Solve 103^2+10t^2=99^2-97^2

$${ 103 }^{ 2 } +10 { t }^{ 2 } = { 99 }^{ 2 } - { 97 }^{ 2 }$$

Solve for t

$$t=-\frac{\sqrt{102170}i}{10}\approx -0-31.964042298i$$

$$t=\frac{\sqrt{102170}i}{10}\approx 31.964042298i$$

Steps by Finding Square Root

Calculate 103 to the power of 2 and get 10609.

$$10609+10t^{2}=99^{2}-97^{2}$$

Calculate 99 to the power of 2 and get 9801.

$$10609+10t^{2}=9801-97^{2}$$

Calculate 97 to the power of 2 and get 9409.

$$10609+10t^{2}=9801-9409$$

Subtract 9409 from 9801 to get 392.

$$10609+10t^{2}=392$$

Subtract 10609 from both sides.

$$10t^{2}=392-10609$$

Subtract 10609 from 392 to get -10217.

$$10t^{2}=-10217$$

Divide both sides by 10.

$$t^{2}=-\frac{10217}{10}$$

The equation is now solved.

$$t=\frac{\sqrt{102170}i}{10} t=-\frac{\sqrt{102170}i}{10}$$

Steps Using the Quadratic Formula

Calculate 103 to the power of 2 and get 10609.

$$10609+10t^{2}=99^{2}-97^{2}$$

Calculate 99 to the power of 2 and get 9801.

$$10609+10t^{2}=9801-97^{2}$$

Calculate 97 to the power of 2 and get 9409.

$$10609+10t^{2}=9801-9409$$

Subtract 9409 from 9801 to get 392.

$$10609+10t^{2}=392$$

Subtract 392 from both sides.

$$10609+10t^{2}-392=0$$

Subtract 392 from 10609 to get 10217.

$$10217+10t^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$10t^{2}+10217=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 0 for b, and 10217 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{0±\sqrt{0^{2}-4\times 10\times 10217}}{2\times 10}$$

Square 0.

$$t=\frac{0±\sqrt{-4\times 10\times 10217}}{2\times 10}$$

Multiply -4 times 10.

$$t=\frac{0±\sqrt{-40\times 10217}}{2\times 10}$$

Multiply -40 times 10217.

$$t=\frac{0±\sqrt{-408680}}{2\times 10}$$

Take the square root of -408680.

$$t=\frac{0±2\sqrt{102170}i}{2\times 10}$$

Multiply 2 times 10.

$$t=\frac{0±2\sqrt{102170}i}{20}$$

Now solve the equation t=\frac{0±2\sqrt{102170}i}{20} when ± is plus.

$$t=\frac{\sqrt{102170}i}{10}$$

Now solve the equation t=\frac{0±2\sqrt{102170}i}{20} when ± is minus.

$$t=-\frac{\sqrt{102170}i}{10}$$

The equation is now solved.

$$t=\frac{\sqrt{102170}i}{10} t=-\frac{\sqrt{102170}i}{10}$$