Solve 7x^2-2x-9

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

Factor

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

Solution Steps

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

$$a+b=-2 ab=7\left(-9\right)=-63$$

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.

$$a=-9 b=7$$

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

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

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

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

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

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

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.

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

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.

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

Square -2.

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

Multiply -4 times 7.

$$x=\frac{-\left(-2\right)±\sqrt{4-28\left(-9\right)}}{2\times 7}$$

Multiply -28 times -9.

$$x=\frac{-\left(-2\right)±\sqrt{4+252}}{2\times 7}$$

Add 4 to 252.

$$x=\frac{-\left(-2\right)±\sqrt{256}}{2\times 7}$$

Take the square root of 256.

$$x=\frac{-\left(-2\right)±16}{2\times 7}$$

The opposite of -2 is 2.

$$x=\frac{2±16}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{2±16}{14}$$

Now solve the equation x=\frac{2±16}{14} when ± is plus. Add 2 to 16.

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

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

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

Now solve the equation x=\frac{2±16}{14} when ± is minus. Subtract 16 from 2.

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

Divide -14 by 14.

$$x=-1$$

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 -1 for x_{2}.

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

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

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

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

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

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

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

Evaluate

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