Solve b^2+26b+48

$$b ^ { 2 } + 26 b + 48$$

Factor

$$\left(b+2\right)\left(b+24\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb+48. To find p and q, set up a system to be solved.

$$p+q=26 pq=1\times 48=48$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 48.

$$1,48 2,24 3,16 4,12 6,8$$

Calculate the sum for each pair.

$$1+48=49 2+24=26 3+16=19 4+12=16 6+8=14$$

The solution is the pair that gives sum 26.

$$p=2 q=24$$

Rewrite b^{2}+26b+48 as \left(b^{2}+2b\right)+\left(24b+48\right).

$$\left(b^{2}+2b\right)+\left(24b+48\right)$$

Factor out b in the first and 24 in the second group.

$$b\left(b+2\right)+24\left(b+2\right)$$

Factor out common term b+2 by using distributive property.

$$\left(b+2\right)\left(b+24\right)$$

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.

$$b^{2}+26b+48=0$$

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.

$$b=\frac{-26±\sqrt{26^{2}-4\times 48}}{2}$$

Square 26.

$$b=\frac{-26±\sqrt{676-4\times 48}}{2}$$

Multiply -4 times 48.

$$b=\frac{-26±\sqrt{676-192}}{2}$$

Add 676 to -192.

$$b=\frac{-26±\sqrt{484}}{2}$$

Take the square root of 484.

$$b=\frac{-26±22}{2}$$

Now solve the equation b=\frac{-26±22}{2} when ± is plus. Add -26 to 22.

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

Divide -4 by 2.

$$b=-2$$

Now solve the equation b=\frac{-26±22}{2} when ± is minus. Subtract 22 from -26.

$$b=-\frac{48}{2}$$

Divide -48 by 2.

$$b=-24$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2 for x_{1} and -24 for x_{2}.

$$b^{2}+26b+48=\left(b-\left(-2\right)\right)\left(b-\left(-24\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$b^{2}+26b+48=\left(b+2\right)\left(b+24\right)$$

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.

$$x ^ 2 +26x +48 = 0$$

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

$$r + s = -26 rs = 48$$

Two numbers r and s sum up to -26 exactly when the average of the two numbers is \frac{1}{2}*-26 = -13. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -13 - u s = -13 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 48

$$(-13 - u) (-13 + u) = 48$$

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

$$169 - u^2 = 48$$

Simplify the expression by subtracting 169 on both sides

$$-u^2 = 48-169 = -121$$

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

$$u^2 = 121 u = \pm\sqrt{121} = \pm 11$$

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

$$r =-13 - 11 = -24 s = -13 + 11 = -2$$

Evaluate

$$\left(b+2\right)\left(b+24\right)$$