Solve 45(p^2-1)=56p

$$45 ( p ^ { 2 } - 1 ) = 56 p$$

Solve for p

$$p=-\frac{5}{9}\approx -0.555555556$$

$$p = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 45 by p^{2}-1.

$$45p^{2}-45=56p$$

Subtract 56p from both sides.

$$45p^{2}-45-56p=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$45p^{2}-56p-45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 45p^{2}+ap+bp-45. To find a and b, set up a system to be solved.

$$a+b=-56 ab=45\left(-45\right)=-2025$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -2025.

$$1,-2025 3,-675 5,-405 9,-225 15,-135 25,-81 27,-75 45,-45$$

Calculate the sum for each pair.

$$1-2025=-2024 3-675=-672 5-405=-400 9-225=-216 15-135=-120 25-81=-56 27-75=-48 45-45=0$$

The solution is the pair that gives sum -56.

$$a=-81 b=25$$

Rewrite 45p^{2}-56p-45 as \left(45p^{2}-81p\right)+\left(25p-45\right).

$$\left(45p^{2}-81p\right)+\left(25p-45\right)$$

Factor out 9p in the first and 5 in the second group.

$$9p\left(5p-9\right)+5\left(5p-9\right)$$

Factor out common term 5p-9 by using distributive property.

$$\left(5p-9\right)\left(9p+5\right)$$

To find equation solutions, solve 5p-9=0 and 9p+5=0.

$$p=\frac{9}{5} p=-\frac{5}{9}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 45 by p^{2}-1.

$$45p^{2}-45=56p$$

Subtract 56p from both sides.

$$45p^{2}-45-56p=0$$

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.

$$45p^{2}-56p-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 45 for a, -56 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 45\left(-45\right)}}{2\times 45}$$

Square -56.

$$p=\frac{-\left(-56\right)±\sqrt{3136-4\times 45\left(-45\right)}}{2\times 45}$$

Multiply -4 times 45.

$$p=\frac{-\left(-56\right)±\sqrt{3136-180\left(-45\right)}}{2\times 45}$$

Multiply -180 times -45.

$$p=\frac{-\left(-56\right)±\sqrt{3136+8100}}{2\times 45}$$

Add 3136 to 8100.

$$p=\frac{-\left(-56\right)±\sqrt{11236}}{2\times 45}$$

Take the square root of 11236.

$$p=\frac{-\left(-56\right)±106}{2\times 45}$$

The opposite of -56 is 56.

$$p=\frac{56±106}{2\times 45}$$

Multiply 2 times 45.

$$p=\frac{56±106}{90}$$

Now solve the equation p=\frac{56±106}{90} when ± is plus. Add 56 to 106.

$$p=\frac{162}{90}$$

Reduce the fraction \frac{162}{90} to lowest terms by extracting and canceling out 18.

$$p=\frac{9}{5}$$

Now solve the equation p=\frac{56±106}{90} when ± is minus. Subtract 106 from 56.

$$p=-\frac{50}{90}$$

Reduce the fraction \frac{-50}{90} to lowest terms by extracting and canceling out 10.

$$p=-\frac{5}{9}$$

The equation is now solved.

$$p=\frac{9}{5} p=-\frac{5}{9}$$

Steps for Completing the Square

Use the distributive property to multiply 45 by p^{2}-1.

$$45p^{2}-45=56p$$

Subtract 56p from both sides.

$$45p^{2}-45-56p=0$$

Add 45 to both sides. Anything plus zero gives itself.

$$45p^{2}-56p=45$$

Divide both sides by 45.

$$\frac{45p^{2}-56p}{45}=\frac{45}{45}$$

Dividing by 45 undoes the multiplication by 45.

$$p^{2}-\frac{56}{45}p=\frac{45}{45}$$

Divide 45 by 45.

$$p^{2}-\frac{56}{45}p=1$$

Divide -\frac{56}{45}, the coefficient of the x term, by 2 to get -\frac{28}{45}. Then add the square of -\frac{28}{45} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{56}{45}p+\left(-\frac{28}{45}\right)^{2}=1+\left(-\frac{28}{45}\right)^{2}$$

Square -\frac{28}{45} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{56}{45}p+\frac{784}{2025}=1+\frac{784}{2025}$$

Add 1 to \frac{784}{2025}.

$$p^{2}-\frac{56}{45}p+\frac{784}{2025}=\frac{2809}{2025}$$

Factor p^{2}-\frac{56}{45}p+\frac{784}{2025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{28}{45}\right)^{2}=\frac{2809}{2025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{28}{45}\right)^{2}}=\sqrt{\frac{2809}{2025}}$$

Simplify.

$$p-\frac{28}{45}=\frac{53}{45} p-\frac{28}{45}=-\frac{53}{45}$$

Add \frac{28}{45} to both sides of the equation.

$$p=\frac{9}{5} p=-\frac{5}{9}$$