Solve 7x^2+3x-34

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

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.

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

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.This is achieved by dividing both sides of the equation by 7

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

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

$$r + s = -\frac{3}{7} rs = -\frac{34}{7}$$

Two numbers r and s sum up to -\frac{3}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{3}{7} = -\frac{3}{14}. 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>

$$r = -\frac{3}{14} - u s = -\frac{3}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{34}{7}

$$(-\frac{3}{14} - u) (-\frac{3}{14} + u) = -\frac{34}{7}$$

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

$$\frac{9}{196} - u^2 = -\frac{34}{7}$$

Simplify the expression by subtracting \frac{9}{196} on both sides

$$-u^2 = -\frac{34}{7}-\frac{9}{196} = -\frac{961}{196}$$

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

$$u^2 = \frac{961}{196} u = \pm\sqrt{\frac{961}{196}} = \pm \frac{31}{14}$$

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

$$r =-\frac{3}{14} - \frac{31}{14} = -2.429 s = -\frac{3}{14} + \frac{31}{14} = 2$$

Evaluate

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