Solve 4x^2+85x+441

$$4 x ^ { 2 } + 85 x + 441$$

Factor

$$\left(x+9\right)\left(4x+49\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+441. To find a and b, set up a system to be solved.

$$a+b=85 ab=4\times 441=1764$$

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 1764.

$$1,1764 2,882 3,588 4,441 6,294 7,252 9,196 12,147 14,126 18,98 21,84 28,63 36,49 42,42$$

Calculate the sum for each pair.

$$1+1764=1765 2+882=884 3+588=591 4+441=445 6+294=300 7+252=259 9+196=205 12+147=159 14+126=140 18+98=116 21+84=105 28+63=91 36+49=85 42+42=84$$

The solution is the pair that gives sum 85.

$$a=36 b=49$$

Rewrite 4x^{2}+85x+441 as \left(4x^{2}+36x\right)+\left(49x+441\right).

$$\left(4x^{2}+36x\right)+\left(49x+441\right)$$

Factor out 4x in the first and 49 in the second group.

$$4x\left(x+9\right)+49\left(x+9\right)$$

Factor out common term x+9 by using distributive property.

$$\left(x+9\right)\left(4x+49\right)$$

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.

$$4x^{2}+85x+441=0$$

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.

$$x=\frac{-85±\sqrt{85^{2}-4\times 4\times 441}}{2\times 4}$$

Square 85.

$$x=\frac{-85±\sqrt{7225-4\times 4\times 441}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-85±\sqrt{7225-16\times 441}}{2\times 4}$$

Multiply -16 times 441.

$$x=\frac{-85±\sqrt{7225-7056}}{2\times 4}$$

Add 7225 to -7056.

$$x=\frac{-85±\sqrt{169}}{2\times 4}$$

Take the square root of 169.

$$x=\frac{-85±13}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-85±13}{8}$$

Now solve the equation x=\frac{-85±13}{8} when ± is plus. Add -85 to 13.

$$x=-\frac{72}{8}$$

Divide -72 by 8.

$$x=-9$$

Now solve the equation x=\frac{-85±13}{8} when ± is minus. Subtract 13 from -85.

$$x=-\frac{98}{8}$$

Reduce the fraction \frac{-98}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{49}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -9 for x_{1} and -\frac{49}{4} for x_{2}.

$$4x^{2}+85x+441=4\left(x-\left(-9\right)\right)\left(x-\left(-\frac{49}{4}\right)\right)$$

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

$$4x^{2}+85x+441=4\left(x+9\right)\left(x+\frac{49}{4}\right)$$

Add \frac{49}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}+85x+441=4\left(x+9\right)\times \frac{4x+49}{4}$$

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

$$4x^{2}+85x+441=\left(x+9\right)\left(4x+49\right)$$

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

$$x ^ 2 +\frac{85}{4}x +\frac{441}{4} = 0$$

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

$$r + s = -\frac{85}{4} rs = \frac{441}{4}$$

Two numbers r and s sum up to -\frac{85}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{85}{4} = -\frac{85}{8}. 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>

$$r = -\frac{85}{8} - u s = -\frac{85}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{441}{4}

$$(-\frac{85}{8} - u) (-\frac{85}{8} + u) = \frac{441}{4}$$

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

$$\frac{7225}{64} - u^2 = \frac{441}{4}$$

Simplify the expression by subtracting \frac{7225}{64} on both sides

$$-u^2 = \frac{441}{4}-\frac{7225}{64} = -\frac{169}{64}$$

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

$$u^2 = \frac{169}{64} u = \pm\sqrt{\frac{169}{64}} = \pm \frac{13}{8}$$

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

$$r =-\frac{85}{8} - \frac{13}{8} = -12.250 s = -\frac{85}{8} + \frac{13}{8} = -9$$

Evaluate

$$\left(x+9\right)\left(4x+49\right)$$