Solve -9x^2-2x+7

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

Factor

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

Solution Steps

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 negative, the negative number has greater absolute value than the positive. 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.

$$a=7 b=-9$$

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

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

Factor out -x in the first and -1 in the second group.

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

Factor out common term 9x-7 by using distributive property.

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

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.

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

Square -2.

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

Multiply -4 times -9.

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

Multiply 36 times 7.

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

Add 4 to 252.

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

Take the square root of 256.

$$x=\frac{-\left(-2\right)±16}{2\left(-9\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±16}{2\left(-9\right)}$$

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.

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

Divide 18 by -18.

$$x=-1$$

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

$$r + s = -\frac{2}{9} rs = -\frac{7}{9}$$

Two numbers r and s sum up to -\frac{2}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{2}{9} = -\frac{1}{9}. 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>

$$r = -\frac{1}{9} - u s = -\frac{1}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{9}

$$(-\frac{1}{9} - u) (-\frac{1}{9} + u) = -\frac{7}{9}$$

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

$$\frac{1}{81} - u^2 = -\frac{7}{9}$$

Simplify the expression by subtracting \frac{1}{81} on both sides

$$-u^2 = -\frac{7}{9}-\frac{1}{81} = -\frac{64}{81}$$

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

$$u^2 = \frac{64}{81} u = \pm\sqrt{\frac{64}{81}} = \pm \frac{8}{9}$$

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

$$r =-\frac{1}{9} - \frac{8}{9} = -1 s = -\frac{1}{9} + \frac{8}{9} = 0.778$$

Evaluate

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