Solve 7w^2+42w-189

$$7 w ^ { 2 } + 42 w - 189$$

Factor

$$7\left(w-3\right)\left(w+9\right)$$

Solution Steps

Factor out 7.

$$7\left(w^{2}+6w-27\right)$$

Consider w^{2}+6w-27. Factor the expression by grouping. First, the expression needs to be rewritten as w^{2}+aw+bw-27. To find a and b, set up a system to be solved.

$$a+b=6 ab=1\left(-27\right)=-27$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -27.

$$-1,27 -3,9$$

Calculate the sum for each pair.

$$-1+27=26 -3+9=6$$

The solution is the pair that gives sum 6.

$$a=-3 b=9$$

Rewrite w^{2}+6w-27 as \left(w^{2}-3w\right)+\left(9w-27\right).

$$\left(w^{2}-3w\right)+\left(9w-27\right)$$

Factor out w in the first and 9 in the second group.

$$w\left(w-3\right)+9\left(w-3\right)$$

Factor out common term w-3 by using distributive property.

$$\left(w-3\right)\left(w+9\right)$$

Rewrite the complete factored expression.

$$7\left(w-3\right)\left(w+9\right)$$

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.

$$7w^{2}+42w-189=0$$

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.

$$w=\frac{-42±\sqrt{42^{2}-4\times 7\left(-189\right)}}{2\times 7}$$

Square 42.

$$w=\frac{-42±\sqrt{1764-4\times 7\left(-189\right)}}{2\times 7}$$

Multiply -4 times 7.

$$w=\frac{-42±\sqrt{1764-28\left(-189\right)}}{2\times 7}$$

Multiply -28 times -189.

$$w=\frac{-42±\sqrt{1764+5292}}{2\times 7}$$

Add 1764 to 5292.

$$w=\frac{-42±\sqrt{7056}}{2\times 7}$$

Take the square root of 7056.

$$w=\frac{-42±84}{2\times 7}$$

Multiply 2 times 7.

$$w=\frac{-42±84}{14}$$

Now solve the equation w=\frac{-42±84}{14} when ± is plus. Add -42 to 84.

$$w=\frac{42}{14}$$

Divide 42 by 14.

$$w=3$$

Now solve the equation w=\frac{-42±84}{14} when ± is minus. Subtract 84 from -42.

$$w=-\frac{126}{14}$$

Divide -126 by 14.

$$w=-9$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -9 for x_{2}.

$$7w^{2}+42w-189=7\left(w-3\right)\left(w-\left(-9\right)\right)$$

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

$$7w^{2}+42w-189=7\left(w-3\right)\left(w+9\right)$$

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 7

$$x ^ 2 +6x -27 = 0$$

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

$$r + s = -6 rs = -27$$

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 = -27

$$(-3 - u) (-3 + u) = -27$$

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

$$9 - u^2 = -27$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = -27-9 = -36$$

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

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

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

$$r =-3 - 6 = -9 s = -3 + 6 = 3$$

Evaluate

$$7\left(w-3\right)\left(w+9\right)$$