Solve 56-60b+4b^2=0

$$56-60b+4 { b }^{ 2 } = 0$$

Solve for b

$$b=1$$

$$b=14$$

Steps Using Factoring By Grouping

Divide both sides by 4.

$$14-15b+b^{2}=0$$

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

$$b^{2}-15b+14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as b^{2}+ab+bb+14. To find a and b, set up a system to be solved.

$$a+b=-15 ab=1\times 14=14$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -15.

$$a=-14 b=-1$$

Rewrite b^{2}-15b+14 as \left(b^{2}-14b\right)+\left(-b+14\right).

$$\left(b^{2}-14b\right)+\left(-b+14\right)$$

Factor out b in the first and -1 in the second group.

$$b\left(b-14\right)-\left(b-14\right)$$

Factor out common term b-14 by using distributive property.

$$\left(b-14\right)\left(b-1\right)$$

To find equation solutions, solve b-14=0 and b-1=0.

$$b=14 b=1$$

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.

$$4b^{2}-60b+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -60 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 4\times 56}}{2\times 4}$$

Square -60.

$$b=\frac{-\left(-60\right)±\sqrt{3600-4\times 4\times 56}}{2\times 4}$$

Multiply -4 times 4.

$$b=\frac{-\left(-60\right)±\sqrt{3600-16\times 56}}{2\times 4}$$

Multiply -16 times 56.

$$b=\frac{-\left(-60\right)±\sqrt{3600-896}}{2\times 4}$$

Add 3600 to -896.

$$b=\frac{-\left(-60\right)±\sqrt{2704}}{2\times 4}$$

Take the square root of 2704.

$$b=\frac{-\left(-60\right)±52}{2\times 4}$$

The opposite of -60 is 60.

$$b=\frac{60±52}{2\times 4}$$

Multiply 2 times 4.

$$b=\frac{60±52}{8}$$

Now solve the equation b=\frac{60±52}{8} when ± is plus. Add 60 to 52.

$$b=\frac{112}{8}$$

Divide 112 by 8.

$$b=14$$

Now solve the equation b=\frac{60±52}{8} when ± is minus. Subtract 52 from 60.

$$b=\frac{8}{8}$$

Divide 8 by 8.

$$b=1$$

The equation is now solved.

$$b=14 b=1$$

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.

$$4b^{2}-60b+56=0$$

Subtract 56 from both sides of the equation.

$$4b^{2}-60b+56-56=-56$$

Subtracting 56 from itself leaves 0.

$$4b^{2}-60b=-56$$

Divide both sides by 4.

$$\frac{4b^{2}-60b}{4}=-\frac{56}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$b^{2}+\left(-\frac{60}{4}\right)b=-\frac{56}{4}$$

Divide -60 by 4.

$$b^{2}-15b=-\frac{56}{4}$$

Divide -56 by 4.

$$b^{2}-15b=-14$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}-15b+\left(-\frac{15}{2}\right)^{2}=-14+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-15b+\frac{225}{4}=-14+\frac{225}{4}$$

Add -14 to \frac{225}{4}.

$$b^{2}-15b+\frac{225}{4}=\frac{169}{4}$$

Factor b^{2}-15b+\frac{225}{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}.

$$\left(b-\frac{15}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{15}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$b-\frac{15}{2}=\frac{13}{2} b-\frac{15}{2}=-\frac{13}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$b=14 b=1$$