Solve 5p^2+2=1-7p

$$5 p ^ { 2 } + 2 = 1 - 7 p$$

Solve for p

$$p=\frac{\sqrt{29}-7}{10}\approx -0.161483519$$

$$p=\frac{-\sqrt{29}-7}{10}\approx -1.238516481$$

Steps Using the Quadratic Formula

Subtract 1 from both sides.

$$5p^{2}+2-1=-7p$$

Subtract 1 from 2 to get 1.

$$5p^{2}+1=-7p$$

Add 7p to both sides.

$$5p^{2}+1+7p=0$$

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.

$$5p^{2}+7p+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 7 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-7±\sqrt{7^{2}-4\times 5}}{2\times 5}$$

Square 7.

$$p=\frac{-7±\sqrt{49-4\times 5}}{2\times 5}$$

Multiply -4 times 5.

$$p=\frac{-7±\sqrt{49-20}}{2\times 5}$$

Add 49 to -20.

$$p=\frac{-7±\sqrt{29}}{2\times 5}$$

Multiply 2 times 5.

$$p=\frac{-7±\sqrt{29}}{10}$$

Now solve the equation p=\frac{-7±\sqrt{29}}{10} when ± is plus. Add -7 to \sqrt{29}.

$$p=\frac{\sqrt{29}-7}{10}$$

Now solve the equation p=\frac{-7±\sqrt{29}}{10} when ± is minus. Subtract \sqrt{29} from -7.

$$p=\frac{-\sqrt{29}-7}{10}$$

The equation is now solved.

$$p=\frac{\sqrt{29}-7}{10} p=\frac{-\sqrt{29}-7}{10}$$

Steps for Completing the Square

Add 7p to both sides.

$$5p^{2}+2+7p=1$$

Subtract 2 from both sides.

$$5p^{2}+7p=1-2$$

Subtract 2 from 1 to get -1.

$$5p^{2}+7p=-1$$

Divide both sides by 5.

$$\frac{5p^{2}+7p}{5}=-\frac{1}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$p^{2}+\frac{7}{5}p=-\frac{1}{5}$$

Divide \frac{7}{5}, the coefficient of the x term, by 2 to get \frac{7}{10}. Then add the square of \frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+\frac{7}{5}p+\left(\frac{7}{10}\right)^{2}=-\frac{1}{5}+\left(\frac{7}{10}\right)^{2}$$

Square \frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$p^{2}+\frac{7}{5}p+\frac{49}{100}=-\frac{1}{5}+\frac{49}{100}$$

Add -\frac{1}{5} to \frac{49}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}+\frac{7}{5}p+\frac{49}{100}=\frac{29}{100}$$

Factor p^{2}+\frac{7}{5}p+\frac{49}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+\frac{7}{10}\right)^{2}=\frac{29}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+\frac{7}{10}\right)^{2}}=\sqrt{\frac{29}{100}}$$

Simplify.

$$p+\frac{7}{10}=\frac{\sqrt{29}}{10} p+\frac{7}{10}=-\frac{\sqrt{29}}{10}$$

Subtract \frac{7}{10} from both sides of the equation.

$$p=\frac{\sqrt{29}-7}{10} p=\frac{-\sqrt{29}-7}{10}$$