Solve 35+40x+5x^2

$$35+40x+5 { x }^{ 2 }$$

Factor

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

Solution Steps

Factor out 5.

$$5\left(7+8x+x^{2}\right)$$

Consider 7+8x+x^{2}. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{2}+8x+7$$

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

$$a+b=8 ab=1\times 7=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

Rewrite x^{2}+8x+7 as \left(x^{2}+x\right)+\left(7x+7\right).

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

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

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

Factor out common term x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

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.

$$5x^{2}+40x+35=0$$

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.

$$x=\frac{-40±\sqrt{40^{2}-4\times 5\times 35}}{2\times 5}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\times 5\times 35}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-40±\sqrt{1600-20\times 35}}{2\times 5}$$

Multiply -20 times 35.

$$x=\frac{-40±\sqrt{1600-700}}{2\times 5}$$

Add 1600 to -700.

$$x=\frac{-40±\sqrt{900}}{2\times 5}$$

Take the square root of 900.

$$x=\frac{-40±30}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-40±30}{10}$$

Now solve the equation x=\frac{-40±30}{10} when ± is plus. Add -40 to 30.

$$x=-\frac{10}{10}$$

Divide -10 by 10.

$$x=-1$$

Now solve the equation x=\frac{-40±30}{10} when ± is minus. Subtract 30 from -40.

$$x=-\frac{70}{10}$$

Divide -70 by 10.

$$x=-7$$

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 -7 for x_{2}.

$$5x^{2}+40x+35=5\left(x-\left(-1\right)\right)\left(x-\left(-7\right)\right)$$

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

$$5x^{2}+40x+35=5\left(x+1\right)\left(x+7\right)$$

Evaluate

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