Solve 4b^2+20b-24

$$4 b ^ { 2 } + 20 b - 24$$

Factor

$$4\left(b-1\right)\left(b+6\right)$$

Solution Steps

Factor out 4.

$$4\left(b^{2}+5b-6\right)$$

Consider b^{2}+5b-6. Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb-6. To find p and q, set up a system to be solved.

$$p+q=5 pq=1\left(-6\right)=-6$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$p=-1 q=6$$

Rewrite b^{2}+5b-6 as \left(b^{2}-b\right)+\left(6b-6\right).

$$\left(b^{2}-b\right)+\left(6b-6\right)$$

Factor out b in the first and 6 in the second group.

$$b\left(b-1\right)+6\left(b-1\right)$$

Factor out common term b-1 by using distributive property.

$$\left(b-1\right)\left(b+6\right)$$

Rewrite the complete factored expression.

$$4\left(b-1\right)\left(b+6\right)$$

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.

$$4b^{2}+20b-24=0$$

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.

$$b=\frac{-20±\sqrt{20^{2}-4\times 4\left(-24\right)}}{2\times 4}$$

Square 20.

$$b=\frac{-20±\sqrt{400-4\times 4\left(-24\right)}}{2\times 4}$$

Multiply -4 times 4.

$$b=\frac{-20±\sqrt{400-16\left(-24\right)}}{2\times 4}$$

Multiply -16 times -24.

$$b=\frac{-20±\sqrt{400+384}}{2\times 4}$$

Add 400 to 384.

$$b=\frac{-20±\sqrt{784}}{2\times 4}$$

Take the square root of 784.

$$b=\frac{-20±28}{2\times 4}$$

Multiply 2 times 4.

$$b=\frac{-20±28}{8}$$

Now solve the equation b=\frac{-20±28}{8} when ± is plus. Add -20 to 28.

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

Divide 8 by 8.

$$b=1$$

Now solve the equation b=\frac{-20±28}{8} when ± is minus. Subtract 28 from -20.

$$b=-\frac{48}{8}$$

Divide -48 by 8.

$$b=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -6 for x_{2}.

$$4b^{2}+20b-24=4\left(b-1\right)\left(b-\left(-6\right)\right)$$

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

$$4b^{2}+20b-24=4\left(b-1\right)\left(b+6\right)$$

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.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 +5x -6 = 0$$

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

$$r + s = -5 rs = -6$$

Two numbers r and s sum up to -5 exactly when the average of the two numbers is \frac{1}{2}*-5 = -\frac{5}{2}. 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>

$$r = -\frac{5}{2} - u s = -\frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -6

$$(-\frac{5}{2} - u) (-\frac{5}{2} + u) = -6$$

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

$$\frac{25}{4} - u^2 = -6$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = -6-\frac{25}{4} = -\frac{49}{4}$$

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

$$u^2 = \frac{49}{4} u = \pm\sqrt{\frac{49}{4}} = \pm \frac{7}{2}$$

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

$$r =-\frac{5}{2} - \frac{7}{2} = -6 s = -\frac{5}{2} + \frac{7}{2} = 1$$

Evaluate

$$4\left(b-1\right)\left(b+6\right)$$