Solve 8.4^2+k^2=14^2

$$8.4 ^ { 2 } + k ^ { 2 } = 14 ^ { 2 }$$

Solve for k

$$k=11.2$$

$$k=-11.2$$

Steps Using Difference of Squares

Calculate 8.4 to the power of 2 and get 70.56.

$$70.56+k^{2}=14^{2}$$

Calculate 14 to the power of 2 and get 196.

$$70.56+k^{2}=196$$

Subtract 196 from both sides.

$$70.56+k^{2}-196=0$$

Subtract 196 from 70.56 to get -125.44.

$$-125.44+k^{2}=0$$

Consider -125.44+k^{2}. Rewrite -125.44+k^{2} as k^{2}-\left(\frac{56}{5}\right)^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(k-\frac{56}{5}\right)\left(k+\frac{56}{5}\right)=0$$

To find equation solutions, solve k-\frac{56}{5}=0 and k+\frac{56}{5}=0.

$$k=\frac{56}{5} k=-\frac{56}{5}$$

Steps by Finding Square Root

Calculate 8.4 to the power of 2 and get 70.56.

$$70.56+k^{2}=14^{2}$$

Calculate 14 to the power of 2 and get 196.

$$70.56+k^{2}=196$$

Subtract 70.56 from both sides.

$$k^{2}=196-70.56$$

Subtract 70.56 from 196 to get 125.44.

$$k^{2}=125.44$$

Take the square root of both sides of the equation.

$$k=\frac{56}{5} k=-\frac{56}{5}$$

Steps Using the Quadratic Formula

Calculate 8.4 to the power of 2 and get 70.56.

$$70.56+k^{2}=14^{2}$$

Calculate 14 to the power of 2 and get 196.

$$70.56+k^{2}=196$$

Subtract 196 from both sides.

$$70.56+k^{2}-196=0$$

Subtract 196 from 70.56 to get -125.44.

$$-125.44+k^{2}=0$$

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.

$$k^{2}-125.44=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -125.44 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{0±\sqrt{0^{2}-4\left(-125.44\right)}}{2}$$

Square 0.

$$k=\frac{0±\sqrt{-4\left(-125.44\right)}}{2}$$

Multiply -4 times -125.44.

$$k=\frac{0±\sqrt{501.76}}{2}$$

Take the square root of 501.76.

$$k=\frac{0±\frac{112}{5}}{2}$$

Now solve the equation k=\frac{0±\frac{112}{5}}{2} when ± is plus.

$$k=\frac{56}{5}$$

Now solve the equation k=\frac{0±\frac{112}{5}}{2} when ± is minus.

$$k=-\frac{56}{5}$$

The equation is now solved.

$$k=\frac{56}{5} k=-\frac{56}{5}$$