Solve p^2+(25-7p)^2=25

$$p ^ { 2 } + ( 25 - 7 p ) ^ { 2 } = 25$$

Solve for p

$$p=3$$

$$p=4$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(25-7p\right)^{2}.

$$p^{2}+625-350p+49p^{2}=25$$

Combine p^{2} and 49p^{2} to get 50p^{2}.

$$50p^{2}+625-350p=25$$

Subtract 25 from both sides.

$$50p^{2}+625-350p-25=0$$

Subtract 25 from 625 to get 600.

$$50p^{2}+600-350p=0$$

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.

$$50p^{2}-350p+600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 50 for a, -350 for b, and 600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-350\right)±\sqrt{\left(-350\right)^{2}-4\times 50\times 600}}{2\times 50}$$

Square -350.

$$p=\frac{-\left(-350\right)±\sqrt{122500-4\times 50\times 600}}{2\times 50}$$

Multiply -4 times 50.

$$p=\frac{-\left(-350\right)±\sqrt{122500-200\times 600}}{2\times 50}$$

Multiply -200 times 600.

$$p=\frac{-\left(-350\right)±\sqrt{122500-120000}}{2\times 50}$$

Add 122500 to -120000.

$$p=\frac{-\left(-350\right)±\sqrt{2500}}{2\times 50}$$

Take the square root of 2500.

$$p=\frac{-\left(-350\right)±50}{2\times 50}$$

The opposite of -350 is 350.

$$p=\frac{350±50}{2\times 50}$$

Multiply 2 times 50.

$$p=\frac{350±50}{100}$$

Now solve the equation p=\frac{350±50}{100} when ± is plus. Add 350 to 50.

$$p=\frac{400}{100}$$

Divide 400 by 100.

$$p=4$$

Now solve the equation p=\frac{350±50}{100} when ± is minus. Subtract 50 from 350.

$$p=\frac{300}{100}$$

Divide 300 by 100.

$$p=3$$

The equation is now solved.

$$p=4 p=3$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(25-7p\right)^{2}.

$$p^{2}+625-350p+49p^{2}=25$$

Combine p^{2} and 49p^{2} to get 50p^{2}.

$$50p^{2}+625-350p=25$$

Subtract 625 from both sides.

$$50p^{2}-350p=25-625$$

Subtract 625 from 25 to get -600.

$$50p^{2}-350p=-600$$

Divide both sides by 50.

$$\frac{50p^{2}-350p}{50}=-\frac{600}{50}$$

Dividing by 50 undoes the multiplication by 50.

$$p^{2}+\left(-\frac{350}{50}\right)p=-\frac{600}{50}$$

Divide -350 by 50.

$$p^{2}-7p=-\frac{600}{50}$$

Divide -600 by 50.

$$p^{2}-7p=-12$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-7p+\left(-\frac{7}{2}\right)^{2}=-12+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-7p+\frac{49}{4}=-12+\frac{49}{4}$$

Add -12 to \frac{49}{4}.

$$p^{2}-7p+\frac{49}{4}=\frac{1}{4}$$

Factor p^{2}-7p+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{7}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$p-\frac{7}{2}=\frac{1}{2} p-\frac{7}{2}=-\frac{1}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$p=4 p=3$$