Solve 4b^2+24b-64

$$4 b ^ { 2 } + 24 b - 64$$

Factor

$$4\left(b-2\right)\left(b+8\right)$$

Solution Steps

Factor out 4.

$$4\left(b^{2}+6b-16\right)$$

Consider b^{2}+6b-16. Factor the expression by grouping. First, the expression needs to be rewritten as b^{2}+pb+qb-16. To find p and q, set up a system to be solved.

$$p+q=6 pq=1\left(-16\right)=-16$$

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 -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 6.

$$p=-2 q=8$$

Rewrite b^{2}+6b-16 as \left(b^{2}-2b\right)+\left(8b-16\right).

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

Factor out b in the first and 8 in the second group.

$$b\left(b-2\right)+8\left(b-2\right)$$

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

$$\left(b-2\right)\left(b+8\right)$$

Rewrite the complete factored expression.

$$4\left(b-2\right)\left(b+8\right)$$

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.

$$4b^{2}+24b-64=0$$

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.

$$b=\frac{-24±\sqrt{24^{2}-4\times 4\left(-64\right)}}{2\times 4}$$

Square 24.

$$b=\frac{-24±\sqrt{576-4\times 4\left(-64\right)}}{2\times 4}$$

Multiply -4 times 4.

$$b=\frac{-24±\sqrt{576-16\left(-64\right)}}{2\times 4}$$

Multiply -16 times -64.

$$b=\frac{-24±\sqrt{576+1024}}{2\times 4}$$

Add 576 to 1024.

$$b=\frac{-24±\sqrt{1600}}{2\times 4}$$

Take the square root of 1600.

$$b=\frac{-24±40}{2\times 4}$$

Multiply 2 times 4.

$$b=\frac{-24±40}{8}$$

Now solve the equation b=\frac{-24±40}{8} when ± is plus. Add -24 to 40.

$$b=\frac{16}{8}$$

Divide 16 by 8.

$$b=2$$

Now solve the equation b=\frac{-24±40}{8} when ± is minus. Subtract 40 from -24.

$$b=-\frac{64}{8}$$

Divide -64 by 8.

$$b=-8$$

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 -8 for x_{2}.

$$4b^{2}+24b-64=4\left(b-2\right)\left(b-\left(-8\right)\right)$$

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

$$4b^{2}+24b-64=4\left(b-2\right)\left(b+8\right)$$

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

$$x ^ 2 +6x -16 = 0$$

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

$$r + s = -6 rs = -16$$

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. 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>

$$r = -3 - u s = -3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -16

$$(-3 - u) (-3 + u) = -16$$

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

$$9 - u^2 = -16$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -16-9 = -25$$

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

$$u^2 = 25 u = \pm\sqrt{25} = \pm 5$$

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

$$r =-3 - 5 = -8 s = -3 + 5 = 2$$

Evaluate

$$4\left(b-2\right)\left(b+8\right)$$