Solve 7x^2+3x-34

$$7 { x }^{ 2 } +3x-34$$

Factor

$$\left(x-2\right)\left(7x+17\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-34. To find a and b, set up a system to be solved.

$$a+b=3 ab=7\left(-34\right)=-238$$

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 -238.

$$-1,238 -2,119 -7,34 -14,17$$

Calculate the sum for each pair.

$$-1+238=237 -2+119=117 -7+34=27 -14+17=3$$

The solution is the pair that gives sum 3.

$$a=-14 b=17$$

Rewrite 7x^{2}+3x-34 as \left(7x^{2}-14x\right)+\left(17x-34\right).

$$\left(7x^{2}-14x\right)+\left(17x-34\right)$$

Factor out 7x in the first and 17 in the second group.

$$7x\left(x-2\right)+17\left(x-2\right)$$

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

$$\left(x-2\right)\left(7x+17\right)$$

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.

$$7x^{2}+3x-34=0$$

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.

$$x=\frac{-3±\sqrt{3^{2}-4\times 7\left(-34\right)}}{2\times 7}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 7\left(-34\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-3±\sqrt{9-28\left(-34\right)}}{2\times 7}$$

Multiply -28 times -34.

$$x=\frac{-3±\sqrt{9+952}}{2\times 7}$$

Add 9 to 952.

$$x=\frac{-3±\sqrt{961}}{2\times 7}$$

Take the square root of 961.

$$x=\frac{-3±31}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-3±31}{14}$$

Now solve the equation x=\frac{-3±31}{14} when ± is plus. Add -3 to 31.

$$x=\frac{28}{14}$$

Divide 28 by 14.

$$x=2$$

Now solve the equation x=\frac{-3±31}{14} when ± is minus. Subtract 31 from -3.

$$x=-\frac{34}{14}$$

Reduce the fraction \frac{-34}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{17}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{17}{7} for x_{2}.

$$7x^{2}+3x-34=7\left(x-2\right)\left(x-\left(-\frac{17}{7}\right)\right)$$

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

$$7x^{2}+3x-34=7\left(x-2\right)\left(x+\frac{17}{7}\right)$$

Add \frac{17}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}+3x-34=7\left(x-2\right)\times \frac{7x+17}{7}$$

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

$$7x^{2}+3x-34=\left(x-2\right)\left(7x+17\right)$$

Evaluate

$$\left(x-2\right)\left(7x+17\right)$$