Solve b^2-12b+20=

$$b ^ { 2 } - 12 b + 20 =$$

Factor

$$\left(b-10\right)\left(b-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb+20. To find p and q, set up a system to be solved.

$$p+q=-12 pq=1\times 20=20$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -12.

$$p=-10 q=-2$$

Rewrite b^{2}-12b+20 as \left(b^{2}-10b\right)+\left(-2b+20\right).

$$\left(b^{2}-10b\right)+\left(-2b+20\right)$$

Factor out b in the first and -2 in the second group.

$$b\left(b-10\right)-2\left(b-10\right)$$

Factor out common term b-10 by using distributive property.

$$\left(b-10\right)\left(b-2\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$b^{2}-12b+20=0$$

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.

$$b=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 20}}{2}$$

Square -12.

$$b=\frac{-\left(-12\right)±\sqrt{144-4\times 20}}{2}$$

Multiply -4 times 20.

$$b=\frac{-\left(-12\right)±\sqrt{144-80}}{2}$$

Add 144 to -80.

$$b=\frac{-\left(-12\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$b=\frac{-\left(-12\right)±8}{2}$$

The opposite of -12 is 12.

$$b=\frac{12±8}{2}$$

Now solve the equation b=\frac{12±8}{2} when ± is plus. Add 12 to 8.

$$b=\frac{20}{2}$$

Divide 20 by 2.

$$b=10$$

Now solve the equation b=\frac{12±8}{2} when ± is minus. Subtract 8 from 12.

$$b=\frac{4}{2}$$

Divide 4 by 2.

$$b=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and 2 for x_{2}.

$$b^{2}-12b+20=\left(b-10\right)\left(b-2\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -12x +20 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 12 rs = 20$$

Two numbers r and s sum up to 12 exactly when the average of the two numbers is \frac{1}{2}*12 = 6. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 6 - u s = 6 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 20

$$(6 - u) (6 + u) = 20$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$36 - u^2 = 20$$

Simplify the expression by subtracting 36 on both sides

$$-u^2 = 20-36 = -16$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =6 - 4 = 2 s = 6 + 4 = 10$$

Evaluate

$$\left(b-10\right)\left(b-2\right)$$