Solve 3w^2-11w-70

$$3 w ^ { 2 } - 11 w - 70$$

Factor

$$\left(w-7\right)\left(3w+10\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3w^{2}+aw+bw-70. To find a and b, set up a system to be solved.

$$a+b=-11 ab=3\left(-70\right)=-210$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -210.

$$1,-210 2,-105 3,-70 5,-42 6,-35 7,-30 10,-21 14,-15$$

Calculate the sum for each pair.

$$1-210=-209 2-105=-103 3-70=-67 5-42=-37 6-35=-29 7-30=-23 10-21=-11 14-15=-1$$

The solution is the pair that gives sum -11.

$$a=-21 b=10$$

Rewrite 3w^{2}-11w-70 as \left(3w^{2}-21w\right)+\left(10w-70\right).

$$\left(3w^{2}-21w\right)+\left(10w-70\right)$$

Factor out 3w in the first and 10 in the second group.

$$3w\left(w-7\right)+10\left(w-7\right)$$

Factor out common term w-7 by using distributive property.

$$\left(w-7\right)\left(3w+10\right)$$

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.

$$3w^{2}-11w-70=0$$

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.

$$w=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\left(-70\right)}}{2\times 3}$$

Square -11.

$$w=\frac{-\left(-11\right)±\sqrt{121-4\times 3\left(-70\right)}}{2\times 3}$$

Multiply -4 times 3.

$$w=\frac{-\left(-11\right)±\sqrt{121-12\left(-70\right)}}{2\times 3}$$

Multiply -12 times -70.

$$w=\frac{-\left(-11\right)±\sqrt{121+840}}{2\times 3}$$

Add 121 to 840.

$$w=\frac{-\left(-11\right)±\sqrt{961}}{2\times 3}$$

Take the square root of 961.

$$w=\frac{-\left(-11\right)±31}{2\times 3}$$

The opposite of -11 is 11.

$$w=\frac{11±31}{2\times 3}$$

Multiply 2 times 3.

$$w=\frac{11±31}{6}$$

Now solve the equation w=\frac{11±31}{6} when ± is plus. Add 11 to 31.

$$w=\frac{42}{6}$$

Divide 42 by 6.

$$w=7$$

Now solve the equation w=\frac{11±31}{6} when ± is minus. Subtract 31 from 11.

$$w=-\frac{20}{6}$$

Reduce the fraction \frac{-20}{6} to lowest terms by extracting and canceling out 2.

$$w=-\frac{10}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7 for x_{1} and -\frac{10}{3} for x_{2}.

$$3w^{2}-11w-70=3\left(w-7\right)\left(w-\left(-\frac{10}{3}\right)\right)$$

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

$$3w^{2}-11w-70=3\left(w-7\right)\left(w+\frac{10}{3}\right)$$

Add \frac{10}{3} to w by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3w^{2}-11w-70=3\left(w-7\right)\times \frac{3w+10}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3w^{2}-11w-70=\left(w-7\right)\left(3w+10\right)$$

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 3

$$x ^ 2 -\frac{11}{3}x -\frac{70}{3} = 0$$

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

$$r + s = \frac{11}{3} rs = -\frac{70}{3}$$

Two numbers r and s sum up to \frac{11}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{11}{3} = \frac{11}{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>

$$r = \frac{11}{6} - u s = \frac{11}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{70}{3}

$$(\frac{11}{6} - u) (\frac{11}{6} + u) = -\frac{70}{3}$$

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

$$\frac{121}{36} - u^2 = -\frac{70}{3}$$

Simplify the expression by subtracting \frac{121}{36} on both sides

$$-u^2 = -\frac{70}{3}-\frac{121}{36} = -\frac{961}{36}$$

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

$$u^2 = \frac{961}{36} u = \pm\sqrt{\frac{961}{36}} = \pm \frac{31}{6}$$

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

$$r =\frac{11}{6} - \frac{31}{6} = -3.333 s = \frac{11}{6} + \frac{31}{6} = 7.000$$

Evaluate

$$\left(w-7\right)\left(3w+10\right)$$