Solve 4x^2+28x+17

$$4 x ^ { 2 } + 28 x + 17$$

Factor

$$4\left(x-\left(-2\sqrt{2}-\frac{7}{2}\right)\right)\left(x-\left(2\sqrt{2}-\frac{7}{2}\right)\right)$$

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.

$$4x^{2}+28x+17=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\times 4\times 17}}{2\times 4}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 4\times 17}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-28±\sqrt{784-16\times 17}}{2\times 4}$$

Multiply -16 times 17.

$$x=\frac{-28±\sqrt{784-272}}{2\times 4}$$

Add 784 to -272.

$$x=\frac{-28±\sqrt{512}}{2\times 4}$$

Take the square root of 512.

$$x=\frac{-28±16\sqrt{2}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-28±16\sqrt{2}}{8}$$

Now solve the equation x=\frac{-28±16\sqrt{2}}{8} when ± is plus. Add -28 to 16\sqrt{2}.

$$x=\frac{16\sqrt{2}-28}{8}$$

Divide -28+16\sqrt{2} by 8.

$$x=2\sqrt{2}-\frac{7}{2}$$

Now solve the equation x=\frac{-28±16\sqrt{2}}{8} when ± is minus. Subtract 16\sqrt{2} from -28.

$$x=\frac{-16\sqrt{2}-28}{8}$$

Divide -28-16\sqrt{2} by 8.

$$x=-2\sqrt{2}-\frac{7}{2}$$

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

$$4x^{2}+28x+17=4\left(x-\left(2\sqrt{2}-\frac{7}{2}\right)\right)\left(x-\left(-2\sqrt{2}-\frac{7}{2}\right)\right)$$

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

$$x ^ 2 +7x +\frac{17}{4} = 0$$

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

$$r + s = -7 rs = \frac{17}{4}$$

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

$$r = -\frac{7}{2} - u s = -\frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{17}{4}

$$(-\frac{7}{2} - u) (-\frac{7}{2} + u) = \frac{17}{4}$$

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

$$\frac{49}{4} - u^2 = \frac{17}{4}$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = \frac{17}{4}-\frac{49}{4} = -8$$

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

$$u^2 = 8 u = \pm\sqrt{8} = \pm \sqrt{8}$$

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

$$r =-\frac{7}{2} - \sqrt{8} = -6.328 s = -\frac{7}{2} + \sqrt{8} = -0.672$$

Evaluate

$$4x^{2}+28x+17$$