Solve 7-9x^2+2x

$$7 - 9 x ^ { 2 } + 2 x$$

Factor

$$\left(1-x\right)\left(9x+7\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-9x^{2}+2x+7$$

Factor the expression by grouping. First, the expression needs to be rewritten as -9x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=2 ab=-9\times 7=-63$$

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 -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=9 b=-7$$

Rewrite -9x^{2}+2x+7 as \left(-9x^{2}+9x\right)+\left(-7x+7\right).

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

Factor out 9x in the first and 7 in the second group.

$$9x\left(-x+1\right)+7\left(-x+1\right)$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(9x+7\right)$$

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.

$$-9x^{2}+2x+7=0$$

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.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)\times 7}}{2\left(-9\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-9\right)\times 7}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-2±\sqrt{4+36\times 7}}{2\left(-9\right)}$$

Multiply 36 times 7.

$$x=\frac{-2±\sqrt{4+252}}{2\left(-9\right)}$$

Add 4 to 252.

$$x=\frac{-2±\sqrt{256}}{2\left(-9\right)}$$

Take the square root of 256.

$$x=\frac{-2±16}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-2±16}{-18}$$

Now solve the equation x=\frac{-2±16}{-18} when ± is plus. Add -2 to 16.

$$x=\frac{14}{-18}$$

Reduce the fraction \frac{14}{-18} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{-2±16}{-18} when ± is minus. Subtract 16 from -2.

$$x=-\frac{18}{-18}$$

Divide -18 by -18.

$$x=1$$

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

$$-9x^{2}+2x+7=-9\left(x-\left(-\frac{7}{9}\right)\right)\left(x-1\right)$$

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

$$-9x^{2}+2x+7=-9\left(x+\frac{7}{9}\right)\left(x-1\right)$$

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

$$-9x^{2}+2x+7=-9\times \frac{-9x-7}{-9}\left(x-1\right)$$

Cancel out 9, the greatest common factor in -9 and 9.

$$-9x^{2}+2x+7=\left(-9x-7\right)\left(x-1\right)$$

Evaluate

$$\left(1-x\right)\left(9x+7\right)$$