Solve 5+34x-7x^2

$$5 + 34 x - 7 x ^ { 2 }$$

Factor

$$\left(5-x\right)\left(7x+1\right)$$

Solution Steps

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

$$-7x^{2}+34x+5$$

Factor the expression by grouping. First, the expression needs to be rewritten as -7x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=34 ab=-7\times 5=-35$$

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 -35.

$$-1,35 -5,7$$

Calculate the sum for each pair.

$$-1+35=34 -5+7=2$$

The solution is the pair that gives sum 34.

$$a=35 b=-1$$

Rewrite -7x^{2}+34x+5 as \left(-7x^{2}+35x\right)+\left(-x+5\right).

$$\left(-7x^{2}+35x\right)+\left(-x+5\right)$$

Factor out 7x in -7x^{2}+35x.

$$7x\left(-x+5\right)-x+5$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(7x+1\right)$$

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.

$$-7x^{2}+34x+5=0$$

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.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-7\right)\times 5}}{2\left(-7\right)}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-7\right)\times 5}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-34±\sqrt{1156+28\times 5}}{2\left(-7\right)}$$

Multiply 28 times 5.

$$x=\frac{-34±\sqrt{1156+140}}{2\left(-7\right)}$$

Add 1156 to 140.

$$x=\frac{-34±\sqrt{1296}}{2\left(-7\right)}$$

Take the square root of 1296.

$$x=\frac{-34±36}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-34±36}{-14}$$

Now solve the equation x=\frac{-34±36}{-14} when ± is plus. Add -34 to 36.

$$x=\frac{2}{-14}$$

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

$$x=-\frac{1}{7}$$

Now solve the equation x=\frac{-34±36}{-14} when ± is minus. Subtract 36 from -34.

$$x=-\frac{70}{-14}$$

Divide -70 by -14.

$$x=5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{7} for x_{1} and 5 for x_{2}.

$$-7x^{2}+34x+5=-7\left(x-\left(-\frac{1}{7}\right)\right)\left(x-5\right)$$

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

$$-7x^{2}+34x+5=-7\left(x+\frac{1}{7}\right)\left(x-5\right)$$

Add \frac{1}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-7x^{2}+34x+5=-7\times \frac{-7x-1}{-7}\left(x-5\right)$$

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

$$-7x^{2}+34x+5=\left(-7x-1\right)\left(x-5\right)$$

Evaluate

$$\left(5-x\right)\left(7x+1\right)$$