Solve -16b+b^2=225

$$- 16 b + b ^ { 2 } = 225$$

Solve for b

$$b=-9$$

$$b=25$$

Steps Using Factoring

Subtract 225 from both sides.

$$-16b+b^{2}-225=0$$

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

$$b^{2}-16b-225=0$$

To solve the equation, factor b^{2}-16b-225 using formula b^{2}+\left(a+b\right)b+ab=\left(b+a\right)\left(b+b\right). To find a and b, set up a system to be solved.

$$a+b=-16 ab=-225$$

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 -225.

$$1,-225 3,-75 5,-45 9,-25 15,-15$$

Calculate the sum for each pair.

$$1-225=-224 3-75=-72 5-45=-40 9-25=-16 15-15=0$$

The solution is the pair that gives sum -16.

$$a=-25 b=9$$

Rewrite factored expression \left(b+a\right)\left(b+b\right) using the obtained values.

$$\left(b-25\right)\left(b+9\right)$$

To find equation solutions, solve b-25=0 and b+9=0.

$$b=25 b=-9$$

Steps Using Factoring By Grouping

Subtract 225 from both sides.

$$-16b+b^{2}-225=0$$

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

$$b^{2}-16b-225=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb-225. To find a and b, set up a system to be solved.

$$a+b=-16 ab=1\left(-225\right)=-225$$

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 -225.

$$1,-225 3,-75 5,-45 9,-25 15,-15$$

Calculate the sum for each pair.

$$1-225=-224 3-75=-72 5-45=-40 9-25=-16 15-15=0$$

The solution is the pair that gives sum -16.

$$a=-25 b=9$$

Rewrite b^{2}-16b-225 as \left(b^{2}-25b\right)+\left(9b-225\right).

$$\left(b^{2}-25b\right)+\left(9b-225\right)$$

Factor out b in the first and 9 in the second group.

$$b\left(b-25\right)+9\left(b-25\right)$$

Factor out common term b-25 by using distributive property.

$$\left(b-25\right)\left(b+9\right)$$

To find equation solutions, solve b-25=0 and b+9=0.

$$b=25 b=-9$$

Steps Using the Quadratic Formula

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.

$$b^{2}-16b=225$$

Subtract 225 from both sides of the equation.

$$b^{2}-16b-225=225-225$$

Subtracting 225 from itself leaves 0.

$$b^{2}-16b-225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16 for b, and -225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-225\right)}}{2}$$

Square -16.

$$b=\frac{-\left(-16\right)±\sqrt{256-4\left(-225\right)}}{2}$$

Multiply -4 times -225.

$$b=\frac{-\left(-16\right)±\sqrt{256+900}}{2}$$

Add 256 to 900.

$$b=\frac{-\left(-16\right)±\sqrt{1156}}{2}$$

Take the square root of 1156.

$$b=\frac{-\left(-16\right)±34}{2}$$

The opposite of -16 is 16.

$$b=\frac{16±34}{2}$$

Now solve the equation b=\frac{16±34}{2} when ± is plus. Add 16 to 34.

$$b=\frac{50}{2}$$

Divide 50 by 2.

$$b=25$$

Now solve the equation b=\frac{16±34}{2} when ± is minus. Subtract 34 from 16.

$$b=-\frac{18}{2}$$

Divide -18 by 2.

$$b=-9$$

The equation is now solved.

$$b=25 b=-9$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$b^{2}-16b=225$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-16b+\left(-8\right)^{2}=225+\left(-8\right)^{2}$$

Square -8.

$$b^{2}-16b+64=225+64$$

Add 225 to 64.

$$b^{2}-16b+64=289$$

Factor b^{2}-16b+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-8\right)^{2}=289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-8\right)^{2}}=\sqrt{289}$$

Simplify.

$$b-8=17 b-8=-17$$

Add 8 to both sides of the equation.

$$b=25 b=-9$$