Solve -54x-42x^2

$$- 54 x - 42 x ^ { 2 }$$

Factor

$$-6x\left(7x+9\right)$$

Solution Steps

Factor out 6.

$$6\left(-9x-7x^{2}\right)$$

Consider -9x-7x^{2}. Factor out x.

$$x\left(-9-7x\right)$$

Rewrite the complete factored expression.

$$6x\left(-7x-9\right)$$

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.

$$-42x^{2}-54x=0$$

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.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}}}{2\left(-42\right)}$$

Take the square root of \left(-54\right)^{2}.

$$x=\frac{-\left(-54\right)±54}{2\left(-42\right)}$$

The opposite of -54 is 54.

$$x=\frac{54±54}{2\left(-42\right)}$$

Multiply 2 times -42.

$$x=\frac{54±54}{-84}$$

Now solve the equation x=\frac{54±54}{-84} when ± is plus. Add 54 to 54.

$$x=\frac{108}{-84}$$

Reduce the fraction \frac{108}{-84} to lowest terms by extracting and canceling out 12.

$$x=-\frac{9}{7}$$

Now solve the equation x=\frac{54±54}{-84} when ± is minus. Subtract 54 from 54.

$$x=\frac{0}{-84}$$

Divide 0 by -84.

$$x=0$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{9}{7} for x_{1} and 0 for x_{2}.

$$-42x^{2}-54x=-42\left(x-\left(-\frac{9}{7}\right)\right)x$$

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

$$-42x^{2}-54x=-42\left(x+\frac{9}{7}\right)x$$

Add \frac{9}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-42x^{2}-54x=-42\times \frac{-7x-9}{-7}x$$

Cancel out 7, the greatest common factor in -42 and -7.

$$-42x^{2}-54x=6\left(-7x-9\right)x$$

Evaluate

$$-6x\left(7x+9\right)$$