Solve 4x^2+34x+52=

$$4 x ^ { 2 } + 34 x + 52 =$$

Factor

$$2\left(x+2\right)\left(2x+13\right)$$

Solution Steps

Factor out 2.

$$2\left(2x^{2}+17x+26\right)$$

Consider 2x^{2}+17x+26. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+26. To find a and b, set up a system to be solved.

$$a+b=17 ab=2\times 26=52$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 52.

$$1,52 2,26 4,13$$

Calculate the sum for each pair.

$$1+52=53 2+26=28 4+13=17$$

The solution is the pair that gives sum 17.

$$a=4 b=13$$

Rewrite 2x^{2}+17x+26 as \left(2x^{2}+4x\right)+\left(13x+26\right).

$$\left(2x^{2}+4x\right)+\left(13x+26\right)$$

Factor out 2x in the first and 13 in the second group.

$$2x\left(x+2\right)+13\left(x+2\right)$$

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

$$\left(x+2\right)\left(2x+13\right)$$

Rewrite the complete factored expression.

$$2\left(x+2\right)\left(2x+13\right)$$

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.

$$4x^{2}+34x+52=0$$

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.

$$x=\frac{-34±\sqrt{34^{2}-4\times 4\times 52}}{2\times 4}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\times 4\times 52}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-34±\sqrt{1156-16\times 52}}{2\times 4}$$

Multiply -16 times 52.

$$x=\frac{-34±\sqrt{1156-832}}{2\times 4}$$

Add 1156 to -832.

$$x=\frac{-34±\sqrt{324}}{2\times 4}$$

Take the square root of 324.

$$x=\frac{-34±18}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-34±18}{8}$$

Now solve the equation x=\frac{-34±18}{8} when ± is plus. Add -34 to 18.

$$x=-\frac{16}{8}$$

Divide -16 by 8.

$$x=-2$$

Now solve the equation x=\frac{-34±18}{8} when ± is minus. Subtract 18 from -34.

$$x=-\frac{52}{8}$$

Reduce the fraction \frac{-52}{8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{13}{2}$$

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{13}{2} for x_{2}.

$$4x^{2}+34x+52=4\left(x-\left(-2\right)\right)\left(x-\left(-\frac{13}{2}\right)\right)$$

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

$$4x^{2}+34x+52=4\left(x+2\right)\left(x+\frac{13}{2}\right)$$

Add \frac{13}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}+34x+52=4\left(x+2\right)\times \frac{2x+13}{2}$$

Cancel out 2, the greatest common factor in 4 and 2.

$$4x^{2}+34x+52=2\left(x+2\right)\left(2x+13\right)$$

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 4

$$x ^ 2 +\frac{17}{2}x +13 = 0$$

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

$$r + s = -\frac{17}{2} rs = 13$$

Two numbers r and s sum up to -\frac{17}{2} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{2} = -\frac{17}{4}. 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>

$$r = -\frac{17}{4} - u s = -\frac{17}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 13

$$(-\frac{17}{4} - u) (-\frac{17}{4} + u) = 13$$

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

$$\frac{289}{16} - u^2 = 13$$

Simplify the expression by subtracting \frac{289}{16} on both sides

$$-u^2 = 13-\frac{289}{16} = -\frac{81}{16}$$

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

$$u^2 = \frac{81}{16} u = \pm\sqrt{\frac{81}{16}} = \pm \frac{9}{4}$$

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

$$r =-\frac{17}{4} - \frac{9}{4} = -6.500 s = -\frac{17}{4} + \frac{9}{4} = -2$$

Evaluate

$$2\left(x+2\right)\left(2x+13\right)$$