Solve 84x^2+70x-294

$$84 { x }^{ 2 } +70x-294$$

Factor

$$14\left(2x-3\right)\left(3x+7\right)$$

Solution Steps

Factor out 14.

$$14\left(6x^{2}+5x-21\right)$$

Consider 6x^{2}+5x-21. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=5 ab=6\left(-21\right)=-126$$

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 -126.

$$-1,126 -2,63 -3,42 -6,21 -7,18 -9,14$$

Calculate the sum for each pair.

$$-1+126=125 -2+63=61 -3+42=39 -6+21=15 -7+18=11 -9+14=5$$

The solution is the pair that gives sum 5.

$$a=-9 b=14$$

Rewrite 6x^{2}+5x-21 as \left(6x^{2}-9x\right)+\left(14x-21\right).

$$\left(6x^{2}-9x\right)+\left(14x-21\right)$$

Factor out 3x in the first and 7 in the second group.

$$3x\left(2x-3\right)+7\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(3x+7\right)$$

Rewrite the complete factored expression.

$$14\left(2x-3\right)\left(3x+7\right)$$

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.

$$84x^{2}+70x-294=0$$

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.

$$x=\frac{-70±\sqrt{70^{2}-4\times 84\left(-294\right)}}{2\times 84}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\times 84\left(-294\right)}}{2\times 84}$$

Multiply -4 times 84.

$$x=\frac{-70±\sqrt{4900-336\left(-294\right)}}{2\times 84}$$

Multiply -336 times -294.

$$x=\frac{-70±\sqrt{4900+98784}}{2\times 84}$$

Add 4900 to 98784.

$$x=\frac{-70±\sqrt{103684}}{2\times 84}$$

Take the square root of 103684.

$$x=\frac{-70±322}{2\times 84}$$

Multiply 2 times 84.

$$x=\frac{-70±322}{168}$$

Now solve the equation x=\frac{-70±322}{168} when ± is plus. Add -70 to 322.

$$x=\frac{252}{168}$$

Reduce the fraction \frac{252}{168} to lowest terms by extracting and canceling out 84.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-70±322}{168} when ± is minus. Subtract 322 from -70.

$$x=-\frac{392}{168}$$

Reduce the fraction \frac{-392}{168} to lowest terms by extracting and canceling out 56.

$$x=-\frac{7}{3}$$

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

$$84x^{2}+70x-294=84\left(x-\frac{3}{2}\right)\left(x-\left(-\frac{7}{3}\right)\right)$$

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

$$84x^{2}+70x-294=84\left(x-\frac{3}{2}\right)\left(x+\frac{7}{3}\right)$$

Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$84x^{2}+70x-294=84\times \frac{2x-3}{2}\left(x+\frac{7}{3}\right)$$

Add \frac{7}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$84x^{2}+70x-294=84\times \frac{2x-3}{2}\times \frac{3x+7}{3}$$

Multiply \frac{2x-3}{2} times \frac{3x+7}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$84x^{2}+70x-294=84\times \frac{\left(2x-3\right)\left(3x+7\right)}{2\times 3}$$

Multiply 2 times 3.

$$84x^{2}+70x-294=84\times \frac{\left(2x-3\right)\left(3x+7\right)}{6}$$

Cancel out 6, the greatest common factor in 84 and 6.

$$84x^{2}+70x-294=14\left(2x-3\right)\left(3x+7\right)$$

Evaluate

$$84x^{2}+70x-294$$